Modelling of Electromagnetic Bandgap Structures using an Alternating Direction Implicit (ADI)/Conformal Finite-Difference Time-Domain Method Wei Song A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy Department of Electronic Engineering c Queen Mary University of London February, 2008
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Modelling of Electromagnetic Bandgap Structuresusing an Alternating Direction Implicit
5.3 Comparison of the memory, the computational time used (on average) in
calculating one k vector in dispersion diagram, and the accuracy of the
results using the Yee’s FDTD and the NFDTD method. . . . . . . . . . . . 184
xviii
Chapter 1
Introduction
1.1 Introduction
Electromagnetic bandgap structures (EBGs) are periodically structured artificial electro-
magnetic media. They generally possess band gaps, a range of frequency in which the
electromagnetic (EM) waves cannot propagate. As the principle of ’bandgap’ applies to
photonics engineering, it is also termed as photonic bandgap (PBG) materials or photonic
crystals (PC). Throughout this thesis, these materials are referenced as EBGs.
EBGs structures have attracted a lot of attention for their versatility in controlling
the propagation of electromagnetic waves [1, 2]. Numerical methods have been used
to predict their performance and assist their design. Among them, the Finite-Difference
Time-Domain (FDTD) method [3] is one of the most popular numerical techniques for
modelling EBGs. As a simple way to discretize the Maxwell’s equations, FDTD does
not require model symmetry or complex mathematical formulation, and hence it can be
applied to model inhomogeneous structures such as EBGs which include defects. As a
straitforward solution to the Maxwell’s equations, FDTD provides accurate temporal re-
sults, which enable the study of EBGs over a wide frequency band. After it was first intro-
duced by Yee[4] in 1966, the FDTD algorithm has been going through continuous modi-
fication, refinements and extensions[5–8], which further enhance the method’s capability
and broaden its appeal. As computer costs keep decline, this versatile method gains more
1
Chapter 1 Introduction 2
popularity and wider range applications in electromagnetic community [9, 10], including
EBGs [11, 12].
In a tremendous amount of FDTD approaches in modelling EBGs, an overwhelming
majority is based on the Yee’s scheme [13–16], using uniform orthogonal meshes. There
is also an alternative FDTD approach developed in the nonorthogonal coordinate system
[17], in which a uniform rhombic grid is employed when modelling the rhombic unit
cell. In that approach, the formulae are derived from the conventional Yee’s scheme with
adjustments for the fixed skewed angle in the grid. However, when the curved unit cell
element is considered, staircasing approximation is employed, either with an orthogonal
grid or with a rhombic grid. It is anticipated that the staircasing approximation will
cause numerical errors when the wavelength of interest is small with regard to the grid
size. Consequently, a dense grid with high spatial resolution is required and this leads to
extensive computation with a large computer memory requirement.
On the other hand, the nonorthogonal FDTD (NFDTD) scheme originated by Holland
in 1983 [5] uses structured conformal meshes when modelling curved structures. Com-
pared to the staircase FDTD scheme, fewer meshes are needed to represent the curved or
oblique boundary of electromagnetic structures. As a result, the computational efficiency
can be greatly improved by using the NFDTD method[18]. This thesis will demonstrate
the computational efficiency of the NFDTD method in modelling EBGs with curved in-
clusions.
However, the NFDTD scheme inherently suffers the late time numerical instability[19–
21]. Since it has been successfully applied to the orthogonal FDTD and removes the
Courant-Friedrich-Levy (CFL) stability condition of FDTD method[7, 22], the alternating
direction implicit (ADI) scheme will be extended to the curvilinear coordinates in this
study and a novel Alternating-Direction Implicit Nonorthogonal Finite-Difference Time-
Domain (ADI-NFDTD) method is proposed. The ADI-NFDTD method has demonstrated
that the time increment (dt) used in the simulation is no longer constrained by the CFL
stability condition. Numerical simulations show that with an increased time increment
(dt), the numerical efficiency of the NFDTD can be improved. Additionally, the numerical
simulations show that the inherent late time instability of the NFDTD method is greatly
Chapter 1 Introduction 3
reduced in the ADI-NFDTD algorithm. The occurrence of the unstable results is delayed
to a much later time. This is quite beneficial in modelling EBGs when wave propagating
with high wave numbers is more prone to suffer the late time instability.
1.2 Research Objective
The objective of this study is to compare the efficiency and accuracy of the nonorthog-
onal FDTD (NFDTD) method and the Yee’s FDTD method when modelling EBGs with
curved inclusions. In light of the fact that the NFDTD suffers the late time instability, an
additional objective is to modify the NFDTD into a more stable, efficient and accurate
method.
1.3 Outline of the Thesis
In Chapter Two, the basics of the EBGs will be briefly introduced. Bloch’s theory and the
dispersion diagrams are presented providing a theoretical insight of EBG structures. This
is followed by a brief review of the numerical methods used for the study of EBGs, in-
cluding the generalized Rayleigh’s method, the Korringa-Kohn-Rostoker (KKR) method,
the Plane Wave Expansion (PWE) method, the Transfer-Matrix Method (TMM) and the
Finite-Difference Time-Domain (FDTD) method. Then EBG configurations and applica-
tions are reviewed.
In Chapter Three, the foundation of FDTD, i.e., the Yee’s algorithm is briefly reviewed.
As far as the mesh generation is concerned, different FDTD grid schemes are presented
in this chapter. Techniques for modelling the EBGs are also presented.
Chapter Four firstly presents the extensions of the Yee’s FDTD method, including the
Chapter 1 Introduction 4
nonorthogonal FDTD (NFDTD) approach, in which a generalized curvilinear coordinate
system is applied. The ADI-FDTD algorithm, in which an alternating-direction implicit
(ADI) method is applied to the orthogonal FDTD method. As the NFDTD suffers late
time instability, ADI scheme is introduced to the general nonorthogonal FDTD method
and the ADI-NFDTD method is proposed. In particular, the formulae of the periodic
boundary condition (PBC) incorporated in the ADI-NFDTD scheme is presented. In the
final part of this chapter, the novel ADI-NFDTD scheme is validated by numerical simu-
lations. The efficiency and the late-time instability improvements are demonstrated.
Then numerical simulation results for the NFDTD and ADI-NFDTD modelling of
electromagnetic bandgap structures are presented and discussed in Chapter Five.
Conclusions are drawn in Chapter Six with the suggestions on future work.
Chapter 2
Basics and an Overview of
Electromagnetic Bandgap Structures
2.1 Introduction
Periodic structures have been studied for many years. In the late 1980s, a fully three-
dimensional periodic structure, operating at microwave frequencies, was realized by
Yablonovitch et al. [1] by mechanically drilling holes into a block of dielectric material.
This material prevented the propagation of EM radiation in any three dimensional spa-
tial direction whereas the material is transparent in its solid form at these wavelengths.
Such artificially engineered periodic materials are generically known as electromagnetic
bandgap (EBG) structures.
The main feature of these materials is the existence of a gap (stopband) in the fre-
quency spectrum of propagating EM waves [2, 23–26]. This bandgap frequency depends
on the permittivity of the dielectric inclusions or background used, the dimensions of the
inclusions/defects, their periodicity and the incidence angle of electromagnetic waves
[27]. This feature leads to a variety of phenomena of both fundamental [28, 29] and practi-
cal [1, 30] interests in the way which fascinated artists and scientists alike. In this chapter,
the basic theory, numerical methods and applications of the EBGs will be reviewed.
5
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 6
2.2 Bloch’s Theorem and the Dispersion Diagram
Symmetry in an electromagnetic structure or system is important to the analysis of wave
behaviour. Most theoretical studies on EBGs are based on an interesting symmetry prop-
erty as the EBGs show periodicity in the dielectric of the material. As a consequence,
Bloch’s theory is developed accordingly to describe the modes in such structures. Based
on Bloch’s theory, a dispersion diagram can be derived to describe the frequency be-
haviour of EBGs. In this section, basic concepts related to the study of EBGs will be
introduced, including: Translational Symmetry, Bloch’s Theorem and Periodic Boundary
Condition (PBC), Brillouin Zones and Dispersion Diagram and Photonic Band Gap.
2.2.1 Translational Symmetry
Definition:
A system with translational symmetry is unchanged by a translation through a displace-
ment d[2].
Suppose we have a function ε(r) in a translationally invariant system. Then, when
we do a translation with an operator Td:
Td[ε(r)] = ε(r + d) = ε(r) (2.1)
A system that has continuous translational symmetry is invariant under the T d’s of
any displacement. An example system which has continuous translational symmetry in
all three directions is free spaceε(r) = 1.
All of the EBGs do not have continuous translational symmetry; instead, they all have
discrete translational symmetry. In other words, they are not translational invariant
under any distance, but only under certain distances - which are a multiple of some fixed
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 7
step length (period). The basic step length is termed as the lattice constant a and the basic
step vector is called the primitive lattice vector a or the fundamental translation vector
a. So, we have ε(r) = ε(r + a) and ε(r) = ε(r + R) where R = sa and s is an integer.
Consequently, these structures can be considered as one unit being repeated over and
over. This unit is known as the unit cell.
Figures 2.1, 2.2 and 2.3 show examples of EBG structures that are periodic in one
dimension, two dimensions and all three dimensions respectively. In figure 2.1, the di-
electric property of the material is repeated in one direction at a period of a. The unit
cell is marked using the blue lines. In figure 2.2, the infinitely long cylindrical rods are
periodically loaded in both x and y direction at a distance of a respectively. The unit cell
is marked using the red dashed lines. The material in figure 2.3 is termed as the woodpile
EBG structure and is also referred to as a layer-by-layer photonic crystal in the physics
literature [31]. A unit cell of this material is shown in figure 2.3(b).
(a)
(b)
Figure 2.1: A part of a one-dimensional EBG structure. This EBG consists of alternating layersof materials of different dielectric properties. These materials are periodically stacked on topof each other. a is the lattice constance. (a) the three-dimensional view of the EBG; (b) theone-dimensional view of the EBG.
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 8
Figure 2.2: (a) A part of the infinitely long two-dimensional EBG structure. (b) Unit cellmarked on the x-y cut plane of the EBG structure.
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 9
(a)
(b)
Figure 2.3: (a) Two angles of view of the woodpile EBG structure. (b) Unit cell for the wood-pile EBG material.
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 10
2.2.2 Bloch’s Theorem and Periodic Boundary Condition
Bloch studied wave propagation in three-dimensionally periodic media in 1928, followed
by an extending theorem in one dimension by Floquet in 1883 [24]. In Bloch’s study, he
proved that waves in such a medium can propagate without scattering, their behaviour
is governed by a periodic envelop function multiplied by a planewave. This means that,
in a photonic crystal with periodic dielectric function:
where ai(i = 1, 2, 3) are three primitive lattice vectors. For a crystal periodic in all three
dimensions, the wave can be expressed in forms of combinations of planewaves and a
periodic function [2]:
H(r) = eik·ru(r) (2.3)
where u is a periodic envelope function.
Equation 2.3 is commonly known as Bloch’s theorem. It is known in solid-state
physics as a Bloch State [32] and in mechanics as a Floquet mode [33].
A simple example is shown in figure 2.4. The structure is repetitive in the y-direction,
and invariant and infinite in the x-direction. So this structure has continuous translational
symmetry in the x-direction and discrete translational symmetry in the y-direction. The
unit cell is highlighted in the figure with a blue box. The primitive lattice vector in this
case is a = ay.
According to Bloch’s theorem, fields in these structures are in Bloch state
H(r) = eikxx·eikyy·uky(y, z), (2.4)
where u(y, z) is a periodic function in y satisfying
u(y + s · a, z) = u(y, z) si = 0,±1,±2, ... . (2.5)
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 11
Figure 2.4: A dielectric configuration with discrete translational symmetry. (a is the latticeconstant.)
According to equation (2.4), this state can be considered as a plane wave, modulated
by a periodic function because of the periodic lattice.
As a result, the field in Bloch’s state can be studied by a so called unit cell approach,
in which only elements in one unit cell are investigated and outside-cell elements are
related and expressed using the following relationship:
H(r + say) = eikxx·eiky(y+sa)·uky(y + sa, z)
= eikysa·(eikxx·eikyy·uky(y, z)
)= eikysa H(r) (2.6)
Equation (2.6) is used as the periodic boundary condition (PBC) in the study of the
infinite EBG structures.
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 12
2.2.3 Brillouin Zone
A key fact about Bloch’s states is that a Bloch state with wave vector ky and a Bloch
state with wave vector ky + mb are identical, where b = 2π/a and m is an integer. That
means the mode frequencies are also periodic in ky : ω(ky) = ω(ky + mb). So we only
need to consider ky to exist in the range −π/a < ky ≤ π/a. This region of important,
nonredundant values of ky is called the Brillouin zone.
Another example shown in figure 2.5 is made up of cylindrical elements periodically
laid in a square lattice. Figure 2.5(a) shows the physical lattice, and the Brillouine Zone
of the reciprocal lattice is plotted in figure 2.5(b). This structure shows not only a discrete
translational symmetries along the x and y directions, but also rotation, mirror-reflection
and inversion symmetry. It is shown [2] that when these symmetries are shown in phys-
ical space, the reciprocal space (k space) shows the same kinds of symmetry. As a result,
in the Brillouin zone shown in figure 2.5(b), every k-point included in it need not to be
considered. Only a smallest region within the Brillouin zone for which the k vector are
not related by symmetry need to be considered. That smallest region is termed as the irre-
ducible Brillouin zone, shown in light blue color; the rest of the Brillouin zone contains
redundant copies of the irreducible zone.
(a) (b)
Figure 2.5: (a) The physical lattice of a EBG made using a square lattice. An arbitrary vectorr is shown. (b) The Brillouin zone of the reciprocal lattice, centered at the origin (Γ). Anarbitrary wave vector k is shown. The irreducible Brillouin zone is the light blue triangularwedge. The special points at the center, corner and face are conventionally known as Γ, Mand X [2].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 13
2.2.4 Dispersion Diagram and Photonic Band Gap
With the knowledge of the irreducible Brillouin zone, the possible modes against the
wave vector k can be plotted. This curve can be plotted in one dimension, two dimen-
sions or three dimensions. These plots provide the dispersion relation and information on
the flow of energy in an intuitive way. Figure 2.6 shows two examples of one-dimensional
diagrams of one-dimensional EBG materials.
Figure 2.6(a) is the dispersion diagram for a uniform dielectric medium, to which a
periodicity of a has been artificially assigned. It is known that in a uniform medium, the
speed of light is reduced by the index of refraction. So the plot is just the light-line given
by
ω(k) =ck√
ε(2.7)
Because the wavevector k repeats itself outside the Brillouin zone, the lines fold back
into the zone when they reach the edges. The dashed lines show the folding effect of
applying Bloch’s theorem with an artificial periodicity a.
In figure 2.6(b), there is a gap in frequency between the upper and lower branches of
the lines - a frequency gap in which no mode, regardless of k, can propagate through the
structure. This gap is called an electromagnetic band gap, which can be further classified
as:
1. Complete Electromagnetic Band Gap: A complete photonic bandgap is a range of ω
in which there are no propagating (real k) solutions of Maxwell’s equations for any k,
surrounded by propagating states above and below the gap.
2. Incomplete Electromagnetic Band Gap: This bandgap only exist over a subset of all
possible wavevectors, polarizations, and/or symmetries.
Figures 2.7 and 2.8 show two-dimensional and three-dimensional dispersion dia-
grams of two two-dimensional EBGs respectively. In this thesis, only the two-dimensional
dispersion diagram is used as an example for numerical validation.
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 14
(a) (b)
Figure 2.6: (a) Dispersion relation (band diagram), frequency ω versus wave number k of auniform one-dimensional medium, (b) Schematic effect on the bands of a physical periodicdielectric variation (inset), where a gap has been opened by splitting the degeneracy at thek = ±π/a. The right-up corner shows the physical lattice and the wave vector. a is the latticeconstant [34].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 15
(a)
(b)
Figure 2.7: (a) A two-dimensional dispersion diagram for a two-dimensional EBGs (illus-trated in figure 2.5), in which a complete bandgap is shown by the light blue color; (b) Atwo-dimensional dispersion diagram where an incomplete bandgap can be found. For ex-ample, the bandgap from normalized frequency 0.404 - 0.552 is only valid with k vector fromΓ to M , and a subset from X to Γ, indicated by the light blue color.
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 16
Figure 2.8: A three-dimensional dispersion diagram for a two-dimensional EBG. The EBGis made of circular rods of radius ρ = 0.6, with optical index of 2.9, lying in vacuum on ahexagonal lattice with period of 4. The horizontal plane gives the Bloch wave vector k. Thevertical axis gives 1/λ. The triangle corresponding to the irreducible Brillouin Zone has beendrawn in the (kx, ky) plane. The parameters about the EBGs can be found in [35].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 17
2.3 An Overview of Numerical Methods for the Modelling of
EBGs
Numerical methods are used to predict the performance of EBGs, either for the pur-
pose of theoretical understanding of EBGs, or for the assistance of EBGs related designs.
These numerical methods include the generalized Rayleigh Identity Method [36–39],
the Korringa-Kohn-Rostoker (KKR) approach [40–42], the Plane Wave Expansion (PWE)
method [43–45], the Transfer Matrix Method (TMM) [46, 47], and the Finite-Difference
Time-Domain (FDTD) method [3], etc. Among them, the Plane Wave Expansion (PWE)
method and the FDTD method are two of the most popular methods. In this section, these
numerical methods will be briefly discussed and the details about the FDTD method will
be presented in the following chapters.
2.3.1 The Generalized Rayleigh’s Identity Method and the Korringa-Kohn-
Rostoker (KKR) Method
Nicorovici and McPhedran et al. extended Rayleigh’s technique from electrostatic to full
electromagnetic problems, for singly[37], doubly[38] and triply[39] periodic systems.
Consider EBGs consisting of an array of cylinders or spheres in an isotropic homoge-
neous dielectric host medium. Denote the fundamental translation vectors of the lattice
as ei(i = 1, 2) for cylinder array or (i = 1, 2, 3) for sphere array. The electromagnetic wave
has the wave number k, the equations for the components of the electric and magnetic
fields decouple and each field component satisfies the Helmholtz equation
(∇2 + k2)f(r) = 0 (2.8)
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 18
The solution f(r) has to fulfill the quasiperiodicity condition
f(r + RP ) = ei ki· RP f(r) ∀p (2.9)
where ki = (ki, θi, ϕi), is the wave vector of the incident radiation and RP stands for the
vectors from the origin of coordinates to the center of the pth cylinder (equation (2.10))
or sphere (equation (2.11)):
RP = p1e1 + p2e2 ≡ (p1, p2) , pi ∈ Z (2.10)
RP = p1e1 + p2e2 + p3e3 ≡ (p1, p2, p3) , pi ∈ Z (2.11)
Green’s function G, which obeys the inhomogeneous Helmholtz equation in the pe-
riodic systems, can be expressed as:
(∇2 + k2)G = −c∑
p
δ(r − RP − ρ)eik0· RP . (2.12)
This satisfies the following quasiperiodicity conditions:
G(r + RP , ρ) = eik0· RP G(r, ρ) (2.13)
and
G(r, ρ + RP ) = e−ik0· RP G(r, ρ) (2.14)
where c = 2π for the cylinder array and c = 1 for the sphere array.
By introducing the lattice sums, Nicorovici et al. obtained a representation of the
Green’s function in terms of a rapidly convergent Neumann series [36–39].
If the solution for the periodic lattice is approximated through a variation-iteration
method instead of being calculated directly, the generalized Rayleigh’s Identity method
becomes the Korringa-Kohn-Rostoker (KKR) method to photonics[39, 48], which is achieved
by Korringa, Kohn and Rostoker [40] using different approaches.
In light of this, the Generalized Rayleigh’s Identity method and the KKR method
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 19
share some common features. The most attractive feature of these two methods is that
the greater part of the work of calculating energy bands consists of the calculation of
certain geometrical ’structure constants’, which need only to be calculated once for each
type of lattice. This leads to a more compact and faster convergent scheme than the PWE
method in the case where the potential V (r) is spherically or cylindrically symmetrical
within the inscribed elements and constant in the space between them. However, the
application of the Generalized Rayleigh’s Identity method and the KKR method is limited
to modelling structures with the spherical or cylindrical symmetry, and for ε(r) being
piecewise constant as well. When the actual potential violates these conditions, these
procedures are not suitable[42].
Compared to the Generalized Rayleigh’s Identity method, the KKR method has even
a limited range of applications. For complex ε(r), the variational KKR method cannot be
used. On the other hand, the Generalized Rayleigh’s Identity method is capable of solv-
ing problems in which the dielectric constant taking on finite or infinite values, or imag-
inary values. The Generalized Rayleigh’s Identity method can also be applied when the
cylinders are composed of an arbitrary number of coaxial circular shells filled with mate-
rials having different complex dielectric constants. However, being variational, the KKR
method can be expected to converge more rapidly within its range of application[48].
2.3.2 Plane Wave Expansion method
Compared with the Generalized Rayleigh’s Identity method or the KKR method, the
Plane Wave Expansion (PWE) is much simpler and the computer code is much easier to
write and runs significantly faster. In terms of convergence, the PWE technique gener-
ally demonstrates slower convergence compared with the two aforementioned methods.
However, there are also numerical examples in which the PWE demonstrates fairly rapid
convergence [42]. Moreover, PWE method is not limited to spherically or cylindrically
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 20
symmetric modulation of the potential. Instead, the PWE method can readily handle
almost all sorts of modulations [42, 49].
In the plane wave expansion method, the work is done in the Fourier space. Denote
k = x1k1 + x2k2 as the two-dimensional wave vector of the wave and
G(h) = h1b1 + h2b2 (2.15)
as a reciprocal-lattice vector. In other word, G is the vector in the Fourier space. The
component of the electric field (or magnetic field) is expanded in plane waves as equation
(2.16).
E(r) =∑
G
B Gei(k+ G)·r (2.16)
where B Gis the Fourier coefficients.
Once the electric field (or magnetic field) is known, the other EM field vectors are
determined.
The inverse dielectric constant is also expanded as:
1ε(r)
=∑
G
κ Gei G·r (2.17)
in which ε is a periodic function in physical space satisfying: ε(r) = ε(r + R); κ G is the
Fourier coefficient which can be written in certain form for certain EM properties and
configuration.
Substituting (2.16) and (2.17) into Maxwell’s equations, the following eigenvalue ex-
pression is obtained: ∑G′
κ G− G′ |k + G|2B G′ =ω2
c2B G . (2.18)
Equation (2.18) can be solved using a standard eigenvalue solver [50, 51].
Unlike some other method (e.g. the KKR method), the PWE method does not rely
on the assumption of spherical or cylindrical symmetry of the structure being modeled
hence it a more general method.
The PWE method is also going through continuous developments which:
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 21
• release the method from restrictions of the complex and frequency-dependent di-
electric function [44, 52];
• and enhance the PWE towards a faster convergence rate [50, 53].
In the original PWE method, the calculation of components characterized by frequency-
dependent, complex dielectric functions is more challenging than calculating purely di-
electric materials, since it requires the solution of a generalized nonlinear eigenvalue
problem. This eigenvalue problem can be solved by a linearization scheme, which re-
quires the diagonalization of an equivalent, enlarged matrix, which means more compu-
tational loads. Kuzmiak et al. reported an alternative PWE approach (as presented in the
previous section) to incorporate the frequency-dependent dielectric function, in which
the generalized eigenvalue problem is reduced to the problem of diagonalizing of a set
of matrices whose size equals the number of plane waves kept in the expansions for the
components of the EM field in the system [44]. Then the PWE method is generalized by
them again in order to handle complex, frequency-dependent dielectric function [52].
The PWE method solves Maxwell equations in the whole region including the inclu-
sion element (region II) and the embedding background (region I). However, a step in
the dielectric constant occurs at the boundary. This makes the expansion of the dielec-
tric constant in (2.17) badly convergent. As a result, huge basis sets may be needed to
accurately describe a photonic band structure [44]. To avoid this problem, an embedding
method is introduced into the PWE method [50, 53] in which the wave equation is only
solved in region I (between the elements), with region II (inside the elements) replaced
by an embedding potential defined on the boundary. Much more rapid convergency is
demonstrated by the use of the embedding method [50, 53].
Although the PWE method is a general method and can treat arbitrary shapes of
the inclusion elements in EBGs, it is more efficient when the shape of element shows
spherical or cylindrical symmetry. Otherwise, large number of basis sets will be required
in the expansion and the PWE method will then become computationally expensive as
the computational time is proportional to the cube of the number of the plane waves [76].
An interest topic of the study of EBGs is the behaviour of impurity modes associated with
the introduction of defects into the EBG structure. While this problem can be tackled
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 22
within a PWE approach using the supercell method in which a single defect is placed
within each supercell of an artificially periodic system, the calculations require a lot of
computer time and memory. Besides, when the EBG is of finite size, it is more difficult to
expand the parameters in the PWE method. On the other hand, the following numerical
method presented is readily to calculate the transmission/reflection coefficients for a EBG
slab with finite thickness.
2.3.3 Modelling EBGs using the Transfer-Matrix Method
Pendry and MacKinnon introduced a complementary technique, i.e. the transfer-matrix
method (TMM), of studying EBG structures in 1992[46]. In the TMM, the total volume of
the system is divided into small cells and the fields in each cell are coupled to those in the
neighboring cells. Then the transfer matrix can be defined by relating the incident fields
on one side of the EBG structure with the outgoing fields on the other side.
Expressing Maxwell’s equations:
∇× E = −∂ B
∂t, ∇× H =
∂ D
∂t(2.19)
in (k, ω) space yields:
k × E = ω B , k × H = −ω D . (2.20)
Assuming that
D = ε E = ε0εrE and B = µ H = µ0µr
H, (2.21)
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 23
equations (2.20) then become equations (2.22) and (2.23):
⎡⎢⎢⎢⎣
x y z
kx ky kz
Ex Ey Ez
⎤⎥⎥⎥⎦ = ωµ
⎡⎢⎢⎢⎣
Hxx
Hyy
Hzz
⎤⎥⎥⎥⎦ (2.22)
⎡⎢⎢⎢⎣
x y z
kx ky kz
Hx Hy Hz
⎤⎥⎥⎥⎦ = −ωε
⎡⎢⎢⎢⎣
Exx
Eyy
Ezz
⎤⎥⎥⎥⎦ (2.23)
Equation (2.22) yields:1
ωµ(kxEy − kyEx) = Hz , (2.24)
and equation (2.23) yields:
kyHz − kzHy = −ωεEx . (2.25)
Substituting Hz in equation (2.25) with the expression in (2.24) yields:
ky[1
ωµ(kxEy − kyEx)] − kzHy = −ωεEx (2.26)
By making the substitution:
H ′ =i
aωε0H (2.27)
then equation (2.28) can be obtained:
(iakz)H ′y =
iakyc2µ−1
r
a2ω2[(iakx)Ey − (iaky)Ex] − εrEx . (2.28)
Applying the simple cubic mesh, the fields are defined by vectors a, b, c of length a in the
x, y, z directions, respectively. Transforming equation (2.28) back into real space yields
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 24
equation (2.29) from which the z components of the vectors are eliminated:
H ′y(r + c) = −εr(r + c)Ex(r + c) + H ′
y(r)
− c2µ−1r (r − b + c)
a2ω2[Ey(r + a − b + c) − Ey(r − b + c) − Ex(r + c) + Ex(r − b + c)]
+c2µ−1
r (r + c)a2ω2
[Ey(r + a + c) − Ey(r + c) − Ex(r + b + c) + Ex(r + c)]
(2.29)
Another three equations (2.30)- (2.32) can be obtained in a similar way:
H ′x(r + c) = −εr(r + c)Ey(r + c) + H ′
x(r)
− c2µ−1r (r − a + c)
a2ω2[Ey(r + c) − Ey(r − a + c) − Ex(r − a + b + c) + Ex(r − a + c)]
+c2µ−1
r (r + c)a2ω2
[Ey(r + a + c) − Ey(r + c) − Ex(r + b + c) + Ex(r + c)] (2.30)
Ex(r + c) = +a2ω2
c2µr(r)H ′
y(r) + Ex(r)
+ ε−1r (r)[H ′
y(r − a) − H ′y(r) − H ′
x(r − b) + H ′x(r)]
− ε−1r (r)[H ′
y(r) − H ′y(r + a) − H ′
x(r + a − b) + H ′x(r + a)] (2.31)
Ey(r + c) = −a2ω2
c2µr(r)H ′
x(r) + Ey(r)
+ ε−1r (r)[H ′
y(r − a) − H ′y(r) − H ′
x(r − b) + H ′x(r)]
− ε−1r (r + b)[H ′
y(r − a + b) − H ′y(r + b) − H ′
x(r) + H ′x(r + b)] (2.32)
Equations (2.29) and (2.30) express the H fields on the next plane of cells in terms of
the E fields on the same plane, and the H fields on the previous plane. Equations (2.31)
and (2.32) express the E fields on the next plane in terms of the E and H fields on the
previous plane. Thus, given the x, y components of the E and H fields on one side of
a dielectric structure, the x, y components of the E and H fields on the other side can
be found by integrating through out the structure. For a dielectric structure containing
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 25
(L × L × L) cells, the dimensions of the transfer matrix are 4L2.
The TMM has the advantage that the transmission coefficients and attenuation coeffi-
cients for incident electromagnetic waves of various frequencies can be obtained directly
from the calculations. This method can also be efficiently used in cases when the PWE
method fails or becomes too time consuming. In particular, when the dielectric func-
tion ε is frequency dependent, or when ε has large imaginary values, Fourier expansion
methods are not efficient [47]. However, disordered systems and periodic systems with
imperfections can be easily studied by the TMM method.
Using the TMM, the band property of an infinite periodic system can be calculated,
but the main advantage of this method is the calculation of the transmission and reflec-
tion coefficients for EM waves of various frequencies incident on a finite-thickness slab
of the EBG material. In that case, the material is assumed to be periodic in the directions
parallel to the interfaces.
On the other hand, the Finite-Different Time-Domain (FDTD) method is more flexible
to model arbitrary shaped configurations and complicated dielectric properties in finite
or infinite structures. The computational effort in the FDTD method is proportional to
the number of the representative points on the spatial mesh. In the following topic, the
applications of the FDTD method in modelling the EBGs is briefly reviewed. Details
about the FDTD method and its extensions will be presented in the next two Chapters.
2.3.4 Modelling EBGs using the Finite-Different Time-Domain Method
The FDTD method is one of the most popular numerical methods for the solution of prob-
lems in electromagnetics. As a simple way to discretize Maxwell’s equations, FDTD does
not require model symmetry or complex mathematical formulation. As a straitforward
solution to the Maxwell’s equations, it provides accurate temporal results, which enable
the study of wide frequency range problems. The first FDTD algorithm was introduced
by Yee[4] in 1966. Since then, modification and extensions of this method [5–7, 54–56]
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 26
have been going through continuously.
The FDTD approach has several advantages for modeling EBG structures. It can read-
ily incorporate the very complex geometries associated with the EBG structures them-
selves and their integration with other devices. It can deal with a variety of complex
materials. It also allows the investigations of the broad and narrow bandwidth responses
of an EBG structure from a single simulation. There are over 3600 papers published by
September 2007, reporting EBGs related research by means of FDTD simulations, and
approximately 79% of them are published after the year 2000. A small fraction of these
researches is reviewed in the following section.
• Kesler et al. reported an antenna design with the use of EBG structures as planar
reflectors in 1996 [57]. Field patterns calculated using the FDTD method are in good
agreement with the measured patterns.
• Qian, Radisic and Itoh reported the investigation of four types of EBG structures as
synthesized dielectric materials which possess distinctive stopbands for microstrip lines
experimentally and by simulations using FDTD in 1997 and found an agreement[58].
• Boroditsky, Coccioli and Yablonovitch analyzed the dispersion diagram of an EBG
consisting of a perforated dielectric slab and the properties of a micro-cavity formed by
introducing a defect into such a crystal using the FDTD method in 1998 [59].
• Chutinan and Noda studied the waveguide created by either removing one stripe
from a three-dimensional woodpile EBG or filling up or decreasing the sizes of air holes
from a two-dimensional EBG slab using the FDTD method in 1999 [60] and 2000 [61].
• Yang and Rahmat-Samii analyzed the mushroom-like EBGs using the FDTD simula-
tions in regards to their reflection phase characterizations with applications as a ground
plane for low profile antenna design [62](2001) [63](2003) and lowering mutual coupling
for antenna array [64](2003).
• Ozbay et al. presented a study of the localized coupled-cavity modes in two-dimensional
dielectric EBG with the field patterns and the transmission spectra obtained by the FDTD
simulations in 2002 [65].
• Weily and Bird et al. reported the study of a planar resonator antenna based on a
woodpile EBG material in 2005 [66] and a linear array of EBG Horn sectoral antennas in
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 27
2006 [67].
• Kantartzis et al. presented an analysis of double negative metamaterial-based waveg-
uide and antenna devices based on a three-dimensional ADI-FDTD method in 2007 [68].
• Pinto and Obayya reported the study of an EBG cavities using an improved complex-
enveloped ADI-FDTD method in 2007 [69].
• There have been a number of attempts to verify the perfect lens concept realized by
EBG materials [101] using the FDTD method [70–73]. Zhao et al. studied EBGs with ma-
terial frequency dispersion by means of an auxiliary differential equation (ADE) based
dispersive FDTD methods with averaged permittivity along the material boundaries im-
plemented [74, 75] and with spatial dispersion effects considered [12].
It is worth noting that among the vast application of FDTD in modelling the EBG
related structures, there are three approaches based on nonorthogonal coordinate system:
• The finite difference method developed by Chan et al. [76](1995) and Pendry et al.
[77](1998) [78](1999) is often termed as the Order-N method by its authors. This method
has been applied to EBGs with either pure dielectric inclusions [77] or pure metallic inclu-
sions [78]. However, in their present form, they cannot be applied to some complicated
cases such as an EBG whose inclusions contain both dielectric and metallic components
[17].
• Qiu and He developed an FDTD algorithm that is based on a nonorthogonal coordi-
nate system to study EBGs consisting of a skew lattice in 2000 [17, 79]. This method does
not rely on any assumption on the dielectric property of the material to be modelled and
is ready to tackle complex structures such as combinations of dielectric and metallic com-
ponents in the EBG cell. However, the use of a globally uniformed skew grid imposes
the staircase approximation when a curved surface is modelled.
• Roden and Gedney et al. studied the transmission coefficient of a two-dimensional
EBG at oblique incidence using FDTD algorithm under orthogonal and nonorthogonal
grids [154]. The EBG consists of four dielectric rods in y-direction and infinitely laid rods
in x-direction. A set of auxiliary variables is introduced which implicitly accounts for
the phase shift between corresponding points in different unit cells when a plane-wave
source is obliquely incident.
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 28
2.4 An Overview of EBG Applications
The first attempts of EBG applications were based on EBG substrates that were fabricated
by drilling a periodic pattern of holes in the substrate, or by etching a periodic pattern
of circles in the ground plane [1]. However, nowadays, the regularly generated novel
ideas expand this rapidly developing scientific area with extremely high rate. Enormous
novel designs and wide range applications can be found in microwave and radio engi-
neering, optical circuit designs and optics spectroscopy. The potential of EBGs attracts
researchers from communities which stand aside from electromagnetics like acoustics,
hydrodynamics, mechanics, etc [80] .
When EBGs are applied to antennas as substrate, high impedance ground planes[64,
81, 82], or reflector [57, 83–85], their band-gap features are revealed mainly in two ways:
the in-phase reflection, and the suppression of surface-wave propagation. The in-phase
reflection feature leads to low profile antenna designs [62, 81, 82]. Meanwhile, the feature
of surface-wave suppression helps to improve antenna’s performance such as increasing
the antenna gain and reducing back radiation [86–89].
There is another well-known characteristic of EBGs, which is the ability to support
localized electromagnetic modes inside the frequency gap by introducing defects in the
periodic lattice. This leads to the development of two important group of applications:
the highly-directive antennas [27] and the EBG waveguide. The former group has high
directivity due to the limited angular propagation allowed within the EBG material, in-
cluding EBG resonator/superstrate antennas [35, 66, 90, 91] and EBG cavities [92]. The
devices in the latter group can efficiently transmit electromagnetic waves, even for 90
bands with zero radius of curvature[60, 79], including EBG waveguide [93–96], power
splitters, directional couplers [97, 98], switches [65](and the references therein), and the
EBG filters [99, 100].
There are also applications utilizing the passband of the EBGs, e.g., the subwavelength
imaging canalization which is studied extensively by Belov et al. [80, 101–103] (and the
references therein).
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 29
2.4.1 In-Phase Reflection
Ground planes are used in many antennas. They redirect one-half of the radiation into
the opposite direction, improving the antenna gain, and partially shield objects on the
other side. The reflection phase is of special interest when designing the ground plane
of the antennas. The reflection phase is defined as the phase of the reflected electric field
at the reflecting surface. It is normalized to the phase of the incident electric field at the
reflecting surface.
A perfect electric conductor (PEC) has a 180 reflection phase for a normally incident
plane wave. That means, in a conventional antenna having a flat metal sheet as a ground
plane, if the radiating element is too close to the conductive surface, the 180 out of phase
reflected waves will cancel the radiation waves, resulting in poor radiation efficiency.
This radiation efficiency reduction can also be explained as the image currents generated
by a smooth conducting surface cancel the currents in the antenna if the radiating ele-
ment is too close to the ground plane. This problem is often addressed by including a
quarter-wavelength space between the radiating element and the ground plane, so that
the reflected wave is in-phase with the radiation wave at the radiating element. However,
such a structure then requires a minimum thickness of λ/4 [23].
Ideal perfect magnetic conductor (PMC) ground plane will have a 0 reflection phase
for a normally incident plane wave. However, no natural material has ever been found
to realize the magnetic conducting surface[82].
On the other hand, an EBG structure can be designed to realize a PMC-like surface in a
certain frequency band [15, 23]. Ma et al. demonstrated a magnetic surface realized using
a two-dimension Uniplanar Compact Electromagnetic Bandgap (UC-EBG) structure ex-
perimentally and numerically [104]. The UC-EBG has the merit of easy fabrication, and
most UC-EBGs have been designed by etching a periodic pattern on the ground plane
[105, 106]. Except for a stopband over a wide range of frequency observed, the slow-
wave effect is verified when investigating the propagation characteristics of a UC-EBG
structure in the passband.
However, an EBG surface is more than a PMC surface. The reflection phase of a EBG
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 30
surface varies continuously from 180 to −180 versus frequency, not only 180 for a PEC
surface or 0 for a PMC surface. Yang et al. found through their reflection phase study of
a mushroom type EBG structure, that the EBG ground plane requires a reflection phase
in the range of 90±45 for a low profile wire antenna to obtain a good return loss. Figure
2.11(b) demonstrates an example of the reflection phase of a mushroom-like EBG surface
working as ground plane for a dipole antenna. FDTD method is used in this study.
A finite EBG ground plane with 1λ12GHz × 1λ12GHz size is used in their analysis. This
configuration as shown in figure 2.9(a) is termed as metallo-dielectric electromagnetic
band-gap (MD-EBG) structure and is often referred to as the mushroom-like EBG struc-
ture. The height of the dipole over the top surface of the EBG ground plane is 0.02λ 12GHz
so the overall height of the dipole antenna from the bottom ground plane of the EBG
structure is 0.06λ12GHz . The input impedance is matched to a 50Ω transmission line.
The return loss of the dipole antenna over the EBG ground plane is compared with
those of a dipole antenna over a PEC and PMC ground plane of the same dimension (see
figure 2.9(b)(c)). It is seen that the best return loss of −27dB is achieved by the dipole
antenna over the EBG ground plane. By varying the length of the dipole from 0.26λ12GHz
to 0.60λ12GHz , the return loss changes, which is plotted in figure 2.11(a). The frequency
band of the dipole model is 11.5−16.6 GHz according to−10dB return criteria. It is nearly
the same frequency region (11.3−16GHz) as the reflection phase of the EBG surface varies
from 90 + 45 to 90 − 45 (shown in figure 2.11(b)).
A dipole antenna over a thin grounded high dielectric constant slab can also provide
a similar reflection phase curve against frequency and a similar return loss performance.
However, with the same dimensions with the EBG ground plane, the dielectric constant
of the thin slab need to be increased to εr = 20. More to the point, the EBG antenna
also demonstrates a higher gain and lower back lobe in terms of return loss due to the
suppression of surface waves, which will be discussed in the next section.
More detailed discussion about this EBG antenna can be found in reference [63].
As demonstrated in the above example, with the in-phase reflection, the radiation
element of the antennas can be placed very close to the EBG structure. In this way, many
low profile antenna designs can be realized. A low-profile cavity backed slot antenna on
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 31
(a)
(b)
(c)
Figure 2.9: (a) Geometry of a mushroom-like EBG structure, which consists of a latticeof metal plates, connected to a solid metal sheet by vertical conducting vias. The EBGsis with the following parameters: W = 0.12λ12GHz , g = 0.02λ12GHz , h = 0.04λ12GHz ,r = 0.005λ12GHz, εr = 2.20, where W is the patch width, g is the gap width, h is the substratethickness, r is the radius of the vias, and εr is the substrate permittivity. (b) The Antenna withthe PEC or PMC ground plane. (c) The Antenna with the EBG ground plane [63].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 32
Figure 2.10: FDTD simulated return loss results of the dipole antenna over the EBG, PEC,and PMC ground planes of the same dimension. The dipole hight is 0.04λ12GHz with thePEC and PMC ground plane and the overall antenna height is 0.06λ12GHz [63].
(a) (b)
Figure 2.11: The FDTD result of (a)return loss of the dipole with its length varying from0.26λ12GHz to 0.60λ12GHz ; (b)The reflection phases of the mushroom-like EBG surface versusfrequency. The frequency band of the dipole model is 11.5 − 16.6 GHz according to −10dBreturn criteria. The frequency band of the plane wave model is 11.3 − 16GHz for 90 ± 45
reflection phase region. [63].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 33
a UC-PBG structure was proposed in [81](figure 2.12). The cavity depth of the proposed
antenna (35 mil) is 16 times thinner than that of a conventional λ/4 wavelength cavity
slot antenna (559 mil). Similarly, a low-profile circularly-polarized patch antenna was
proposed in [62] (figure 2.13). The EBG ground plane size can be as small as (0.82λ ×0.82λ). The overall height of the EBG structure in conjunction with the antenna proposed
can be 0.1λ (λ corresponds to the working frequency in free space).
Figure 2.12: Schematic, cross section of the proposed slot antenna loaded with UC-PBG re-flector, and the top view of the UC-PBG. [81].
Figure 2.13: Configuration of a square curl antenna over an EBG surface. The size of theground plane is 1λ×1λ where λ is the free-space wavelength at working frequency 1.57GHz.The low profile curl antenna is with height of 0.06λ. [62].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 34
2.4.2 Suppression of Surface Waves
Another property of metals is that they support surface waves. These are propagating
electromagnetic waves that are bound to the interface between two dissimilar materials,
e.g. metal and free space. At microwave frequencies, they are nothing more than the
normal currents that occur on any electric conductor [23]. The surface waves decay ex-
ponentially into the surrounding materials and will not couple to external plane waves
if the metal surface is smooth and flat [107]. However, the presence of bends, discontinu-
ities or surface texture will result in the surface waves radiating vertically [23].
On a finite ground plane, surface waves propagate until they reach an edge or corner,
where they can radiate into free space. More to the point, if multiple antennas share the
same ground plane, surface currents can cause unwanted mutual coupling.
By incorporating a special texture on a conducting surface, it is possible to alter its EM
properties. In the circumstance where the period of the surface texture is much smaller
than the wavelength, the structure can be described using an effective medium model,
and its qualities can be summarized and expressed by the surface impedance [107]. A
smooth conducting sheet has low surface impedance, but with a specially designed ge-
ometry, a textured surface can have high surface impedance.
The surface impedance of the textured metal surface can be characterized by an equiv-
alent parallel resonant LC circuit. At low frequencies, it is inductive, and supports trans-
verse magnetic (TM) waves. At high frequencies it is capacitive, and supports transverse
electric (TE) waves. Near the LC resonance frequency, the surface impedance is very
high. In this region, waves are not bound onto the surface; instead, they radiate into the
surrounding space [23].
With the suppression of surface waves, the radiation pattern of the antenna can be
improved by the EBG ground plane. Figure 2.14 shows a higher gain and lower back
lobe than the dipole antenna over thin grounded high dielectric constant (εr = 20) slab
with the same dimension.
The suppression of surface wave has been done with several geometries, such as a
metal sheet covered with small bumps [108], or a corrugated metal slab [107, 109] (and
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 35
(a)
(b)
Figure 2.14: Radiation pattern comparison of dipoles near the thin grounded high dielectricconstant slab and the EBG surface. (a) E-plane pattern. (b) H-plane pattern. The patterns arecalculated at the resonant frequency of 13.6 GHz. Since the high dielectric constant substrateis used in the grounded slab and strong surface waves are excited, the dipole on the slabshows a lower gain and higher back lobe [63].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 36
the references therein), or an array of lumped-circuit elements to produce a thin two-
dimensional structure that should generally be described by band structure concepts,
even though the thickness and periodicity are both much smaller than the operating
wavelength [107]. The following shows an example of bumpy surfaces.
Kitson et al. used triangular lattice of bumps, patterned on a glass substrate coated
with a thin film of silver and found that propagation is prohibited in all directions for
modes with designed energy band [108].
The following are the details about their experiment. An excited atom can only relax
via the spontaneous emission of a photon if its energy matches that of an available op-
tical mode. The interaction between light and materials that are periodically modulated
on the scale of the wavelength of light leads to photonic band gaps. In light of these,
if such a gap coincides in energy with the excited state of an atom within the material
then spontaneous emission will be inhibited. In Kitson’s experiment, the surface consists
of hexagonal array of photoresist dots on a glass substrate (figure 2.15) coated with a
thin film of silver. The thin silver layer is used to support the propagation of the surface
plasmon polariton (SPP), which is a non-radiative transverse magnetic (TM) mode that
propagates at the interface between a metal and a dielectric. The repeat distance is 300 nm
and the dots have a radius of around 100 nm. A white light source and a computer con-
trolled spectrometer were used to produce a collimated TM-polarized monochromatic
beam in the wavelength range 400 to 800 nm. Monochromatic light incident on the prism
at a suitable internal angle θ resonantly excites SPPs that propagate on the silver/air in-
terface. This absorbs energy from the beam, reducing the reflectivity. The intensity of the
light reflected from the prism was monitored.
Figure 2.16 plots the reflectivity data recorded as a function of the photon energy and
kx, the component of the photon momentum in the plane of the silver/air interface. This
data is for a propagation direction Ψ of 100 (inset). The light regions represent high re-
flectivity and the dark regions correspond to low reflectivity. The regions of low reflectiv-
ity (dark) are a result of photons that have been absorbed through the resonant excitation
of SPPs. The dispersion curve of the SPP is directly mapped out by the reflectivity graph.
There is a clear gap in the dispersion curve, centered at 1.98 eV.
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 37
Figure 2.15: A scanning electron micrograph of the hexagonal array of dots. The dots arecomposed of photo-resist on a glass substrate. The surface has been coated with a thin filmof silver to support the propagation of the SPP [108].
Figure 2.16: A sample set of reflectivity data recorded as a function of the photon energy andkx, the component of the photon momentum in the plane of the silver/air interface. This datais for a propagation direction Ψ of 100 (inset) [108].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 38
2.4.3 EBGs working in Defect Modes
Although EBG structures are based on periodicity, many interesting applications have
been generated because the defect EBG has the ability of supporting localized electro-
magnetic modes inside the frequency gap. Defects can be realized in many ways, includ-
ing removing segments of elements from a EBG, changing the shape or EM properties of
segments of elements, or replacing a segment of EBG with other materials. In this sec-
tion, examples of EBG superstrate, EBG waveguides, EBG-splitter and coupler and EBG
tunable filter will be presented.
• EBG Superstrates
In contrast to an EBGs being placed below an antenna as a substrate to miniaturize
the antenna and reduce backward radiation, EBGs can also be placed above the antenna
to enhance antenna performance in terms of directivity [35, 90, 110–112].
Figure 2.17 shows an example of a high directive EBG resonator antenna utilizing a
frequency-selective surface (FSS) superstrate designed by Lee et al.[111]. FSSs are chosen
in Lee’s design because of the fact that they are easy to fabricate using the etching pro-
cesses and they can help achieving a more compact EBG antenna design, especially in
terms of the antenna thickness. Figure 2.18 shows the field distribution of the EBG an-
tenna. A substantial enhancement on the directivity of the antenna by utilizing the EBG
superstrate is demonstrated in figure 2.19.
Enoch et al. reported in [35] in 2003 a device that radiates energy into a very narrow
angular range, based on a two dimensional EBG which is made of circular rods of radius
r = 0.6, with optical index nr = 2.9, lying in vacuum. The rods are arranged on a
triangular/hexagonal lattice with period a = 4 (distance between the centers of the rods).
The dispersion diagram of this EBG is presented in figure 2.20.
When the cell of this original EBG is expanded in the y direction, i.e., the vertical spac-
ing between two grids is enlarged from√
3a/2 ≈ 3.46 to 3.9, a new EBG is formed with
dispersion diagram shown in figure 2.21. At the wavelength λ = 7.93 corresponding to
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 39
Figure 2.17: Geometry of a patch antenna with a strip dipole FSS superstrate [111].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 40
Figure 2.18: Field distribution of the EBG antenna [111].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 41
Figure 2.19: Comparison of the radiation patterns of the FSS antenna composite and the patchantenna only. The directivity of the antenna is substantially enhanced [111].
the horizontal plane at the bottom of figure 2.21, the constant-frequency dispersion dia-
gram of the expanded EBG reduces to a small curve (see figure 2.22). As a consequence,
any source embedded in a slice of this expanded EBG will radiate only with a small range
of wavevector k.
By placing the original (unexpanded) EBG which exhibits bandgap at the same wave-
length, backwards radiation is eliminated. In this way, a device with field radiated
in a narrow angular range using almost any excitation is achieved. The half-power
beamwidth is equal to ±4.0(see figure 2.24), in comparison to the ±4.5 achievable by
an aperture having the same width illuminated by a field with constant amplitude and
phase. By increasing the lateral size of the structure and sacrificing the range of possible
working wavelengths, even narrower radiation pattern can be obtained [35].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 42
Figure 2.20: Three dimensional dispersion diagram of the two dimensional EBG with circularrods lying in vacuum in a triangular lattice [35].
Figure 2.21: The dispersion diagram of the expanded EBG [35].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 43
Figure 2.22: Constant-frequency dispersion diagram of the expanded EBG for λ = 7.93 [35].
Figure 2.23: Total field modulus radiated by the structure excited by the wire source at λ =7.93 [35].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 44
Figure 2.24: Polar emission diagram for the structure excited by the wire source at λ = 7.93[35].
• EBG Waveguide, Splitters and Couplers
The efficient guiding and bending of light by integrated photonic devices is important
to the design of optical circuits for technological and optical computing applications.
Conventional dielectric or metallic waveguides have large scattering losses when sharp
bends are introduced. However, EBG studies enable an efficient way of guiding wave
even for sharp bends[60, 94–96].
Ozbay et al. demonstrated a zig-zag coupled cavity waveguide (CCW) formed by
removing consecutive rods from a two-dimensional EBG with rods loaded in free space
in triangular lattice (shown in figure 2.25(a)). A defect band is observed between 0.857ω0
to 0.949ω0. Complete transmission is seen for certain frequencies within the defect band.
Since the defect band shows sharp band edges compared to the EBG edges, it is suggested
in [65] that this property can be used to construct photonic switches by changing the
position of the defect band.
A Y-shaped splitter (shown in figure 2.26) is also presented in paper [65] in order
to demonstrate the splitting of EM power. The splitter consists of one input CCW and
two output CCWs. The input and output waveguide ports contain five and six coupled
cavities, respectively. As seen in figure 2.26(a), the propagating mode inside the input
CCW splits equally into two output CCW ports for all frequencies within the defect band.
The electric field distribution inside the splitter is computed for frequency ω = 0.916ω0
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 45
and is shown in figure 2.26(b).
If a single rod is placed to the left side of the junction of the Y-splitter, as shown in
figure 2.27(b), the splitter structure becomes a photonic switch. Because the symmetry of
the Y-shaped structure is broken, the power at each output waveguide port is drastically
changed. In this way, the amount of power flow into the output ports can be regulated.
(a) (b)
Figure 2.25: (a) Measured (solid line) and calculated (dotted line) transmission spectra of azig-zag CCW waveguide which contains 16 cavities. (b) Calculated field distribution of thezig-zag CCW waveguide. (Reference: figure 5 in [65].)
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 46
(a) (b)
Figure 2.26: (a) Measured (solid line) and calculated (dotted line) transmission spectra of a Y-shaped coupled-cavity based splitter. (b)Calculated power distribution inside the input andoutput waveguide channels of the splitter for frequency ω = 0.916ω0 . (Reference: figure 7 in[65].)
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 47
(a) (b)
Figure 2.27: (a) Measured (solid line) and calculated (dotted line) transmission spectra of acoupled-cavity switching structure. (b)Calculated field pattern clearly indicates that most ofthe power is coupled to the right port. (Reference: figure 8 in [65].)
EBG waveguide can also be realized based on the three-dimensional layer-by-layer
dielectric EBG structures [60, 96]. Figure 2.28(a) shows the experimental setup. The
woodpile EBG constructing the CCW’s consists of square shaped alumina rods hav-
ing a refractive index 3.1 at the microwave frequencies. The dimension of each rod is
0.32cm×0.32cm×15.25cm. The offset between the rods is 1.12cm. The bandgap of the
EBG extends from 10.6 to 12.8GHz. The defect is formed by removing a single rod from
a unit cell of the EBG crystal. The electric-field vector of the incident EM field was par-
allel to the rods of the defect lines. When the defects exists in adjacent unit cells, very
high transmission of the EM wave was observed within a frequency range inside the
bandgap of the otherwise perfect EBG, which is hereinafter referred to as the waveguide
band. Nearly a complete transmission was observed within the waveguide band for a
straight EBG waveguide (figure 2.28(b)) and greater than 90% transmission for an EBG
waveguide with 40 bend (figure 2.28(c)).
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 48
(a)
(b)
(c)
Figure 2.28: (a) Experimental setup for measuring the transmission-amplitude andtransmission-phase spectra of the coupled cavity waveguides (CCW) in 3D photonic crys-tals. (b) A mechanism to guide light through localized defect modes in a woodpile EBGs. (c)Bending of the EM waves around sharp corners [96].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 49
• EBG Tunable Filters
A variety of filtering devices, based on the two-dimensional dielectric and metallic
EBGs containing nematic liquid crystal materials as defect elements or defect layers have
been explored by Kosmidou, et al. [100]. The FDTD simulation revealed that the defect
states originating from the liquid crystal impurities are tunable by the application of a
local static electric field. Narrow mode linewidths, almost 0.2 nm, and tuning ranges in
the order of tens of nanometers, covering in some cases both the C and L bands, can be
achieved with low operating voltages (0 − 4 V). So the proposed devices are rendered
suitable to operate as a spectral filter in modern optical communication systems.
Depicted in figure 2.29 is the defect EBG filter with air voids filled with liquid crystal
as defect elements. It is based on a photonic crystal consisting of a triangular lattice of
infinitely long air cylinders embedded in silicon. The radius of the circular cross section
of the air rods is set to 0.3a, where a is the lattice constant. The relative permittivity of Si
is considered to be εr = 11.4. The defect air voids at the centre of the device are filled with
E7, a typical nematic liquid crystal material characterized by ordinary and extraordinary
refractive indexes equal to 1.49 and 1.66, respectively. The defect area is surrounded by
five periods of the EBG cells, whereas the lateral width (in y− direction) of the device is
presumed to be infinite. The optical axis direction inside the liquid crystal, lying in the
xy plane, is defined by the tilt angle γ which can be altered by means of applying a local
static electric field.
Figure 2.30 and 2.31 show how defect modes can be tuned by changing the number
of defect void rows and by tuning the tilt angle γ in the defect cylinder rows.
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 50
Figure 2.29: Dielectric EBG structure having air-voids filled with liquid crystal as defect ele-ments [100].
Figure 2.30: Transmission coefficient for various numbers of defect cylinder rows when γ =45 [100].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 51
Figure 2.31: Transmission coefficient for various values of the tilt angle γ in the case of twodefect cylinder rows [100].
In figure 2.32, it can be seen that the discrete defect cylinders are replaced by an E7
layer interposed between two blocks of the EBGs. Between the EBG and the E7 layer,
there exist two thin films of indium tin oxide (ITO) with relative refractive index equal to
1.9 and thickness 0.2a, on which electrods are attached in order to provide a local static
voltage across the liquid crystal slab. Figure 2.33 shows how the tilt angle γ in the E7 slab
varies according to the spatial position and to the different values of the applied voltage.
By altering the optical axis orientation inside of the defect slab, the thickness of E7 layer
Lc, or the distance Ls, the effective permittivity will changes, which in turn causes the
change of the position of the defect modes. Figure 2.34 demonstrates how the position
of the defect modes can be tuned by tuning the applied voltage, assuming Lc = 3a and
Ls = 0.567a.
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 52
Figure 2.32: Dielectric EBG structure with a liquid crystal defect layer [100].
Figure 2.33: Director orientation profile across the liquid crystal defect layer for differentvalues of the applied voltage [100].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 53
Figure 2.34: Transmission coefficient versus normalized frequency for various values of theapplied voltage when Lc = 3a, Ls = 0.567a [100].
Another two-dimension EBG tunable filter is based on EBGs with infinite long metal-
lic rods loaded in a background material with low refractive index (Nr = 1.32) in a square
lattice. The radius of each cylinder cross section is set to 0.2a. As illustrated in figure
2.35(a), the E7 layer is placed between two blocks of EBGs, each with four periods of
the EBG cells in the x− direction, and with infinite long assumed in y− direction. Fig-
ure 2.35(b) shows the tuning effect of the defect modes by changing the applied voltage,
assuming Lc = 4a and Ls = a.
More tuning results can be found in paper [100].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 54
(a)
(b)
Figure 2.35: (a) Metallic EBG structure with a liquid crystal defect layer. (b) Transmissioncoefficient versus normalized frequency for various values of the applied voltage when Lc =4a, Ls = a [100].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 55
2.4.4 Subwavelength Imaging from the passband of the EBGs
Resolution of common imaging systems is restricted by the so-called diffraction limit [80].
This effect limits the minimum diameter d of spot of light formed at the focus of a lens,
given as:
d = 1.22λf
a(2.33)
where λ is the wavelength of the light, f is the focal length of the lens, and a is the diam-
eter of the beam of light, or (if the beam is filling the lens) the diameter of the lens. As a
result, even if one could fabricate an imperfection-free optical system, there would still
be a limit to the resolution of an image created by the conventional optical lens. In order
to overcome the diffraction limit, an artificial material (EBGs) with electromagnetic prop-
erties dramatically different from the materials from nature was proposed as a candidate
for a perfect lens and the theoretical possibility of sub-wavelength imaging was demon-
strated by Pendry in his seminal work [101]. Belov et al. experimentally demonstrated
a possibility to channel the near field distribution of a line source with sub-wavelength
details through a EBG crystal. Channelled intensity maximum having radius of λ/10
has been achieved by the use of an electrically dense lattice of capacitively loaded wires
[80, 103].
Figure 2.36 shows the experimental implements of the EBGs composed of capaci-
tively loaded wire medium (CLWM) and the EBG lens. Figure 2.37 shows isofrequency
contours for the frequency region ka = 0.43 ∼ 0.47. The isofrequency contour of the
host material for ka = 0.46 is shown as the small circle around Γ point. The part of the
isofrequency contour for the EBG corresponding to ka = 0.46, and located within the first
Brillouin Zone, is practically flat. This part is perpendicular to the diagonal of the first
Brillouin Zone. Thus, in order to achieve channeling regime, the interfaces of the slab is
oriented orthogonally to (11)-direction as shown in figure 2.36(b).
Figure 2.38 depicts the simulated amplitude and intensity distribution of a line source
working on ka = 0.46 is excited near the interface of the CLWM slab. A clear channel
through the slab can be observed together with a bright spot having a radius of λ/6
(determined from the intensity distribution at level max(intensity)/2) behind the slab.
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 56
An experimental verification of sub-wavelength imaging using CLWM slabs (shown in
figure 2.36(c) ) demonstrated an impressive resolution of λ/10. This lens can be designed
thick, since the required tunnel thickness is not related with the distance to the source.
Application of this CLWM lens being used in the near-field microscopy in the optical
range is suggested in [80], when the needle of a microscope used as a probe can be located
physically far from the tested source.
(a) (b)
(c) (d)
Figure 2.36: (a) A schematic illustration of the EBG structure composed of capacitively loadedwire medium (CLWM). (b) A schematic illustration of the lens formed by the EBG (CLWM).(c) The implemented CLWM EBG sample and the probe used in the measurements. (d) Aschematic illustration of the loaded wires (A piece of it) [103].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 57
Figure 2.37: Isofrequency contours for the CLWM. The numbers correspond to values ofnormalized frequency ka [103].
(a) (b) .
Figure 2.38: Simulated distribution of electric field (a) amplitude and (b) intensity for thesub-wavelength lens formed by the CLWM operating in the canalization regime [103].
Chapter 2 Basics and an Overview of Electromagnetic Bandgap Structures 58
2.5 Summary
The periodic structures are presently one of the most rapidly advancing sectors in the
electromagnetic arena. In this chapter, the basic theory about the EBG structures, the
numerical methods that are popular in modelling EBGs, and examples from the vast
applications of the EBGs are reviewed. These applications include EBG antennas with
EBGs working as substrate or superstrate of the antennas, EBG waveguide, splitters and
couplers, EBG filters and EBG subwavelength imaging channels.
Chapter 3
A Brief Introduction to the
Finite-Difference Time-Domain
Method for Modelling the EBG
Structures
3.1 Introduction
The Finite-Difference Time-Domain (FDTD) Method [3] has been proven to be one of
the most effective numerical methods in the study of EBGs. As a direct solution to the
Maxwell’s equations, FDTD is simple and straightforward to solve complex EBG struc-
tures. As a time domain solution, it is convenient in dealing with the wide frequency
band characteristics of EBGs.
The foundation of FDTD is laid down by Yee in 1966 [4]. Yee chose a geometric rela-
tion for his spatial sampling of the vector components of the electric and magnetic fields
that robustly represent both the differential and integral forms of Maxwell’s equations.
The original Yee’s FDTD algorithm is second-order accurate in both time and space. Nu-
merical dispersion can be kept small by having a sufficient number of grid spaces per
59
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 60
wavelength. This chapter gives out a brief review of the FDTD fundamentals. The tech-
niques of FDTD in EBG structure modelling are also presented.
3.2 Formulations of the Yee’s FDTD algorithm
Yee’s algorithm deals with both electric and magnetic fields in time and space using the
coupled Maxwell’s curl equations rather than solving for the electric field (or the mag-
netic field) alone with a wave equation.
3.2.1 Maxwell’s Equations
Consider a region of space that has no electric or magnetic current sources, but may have
materials that absorb electric or magnetic field energy. The time-dependent Maxwell’s
equations are given in differential and integral form by:
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 61
where σ∗ : equivalent magnetic loss (ohms/meter)
σ : electric conductivity (siemens/meter)
E : electric field, also called the electric flux density (volt/meter)
H : magnetic field strength (ampere/meter)
D : electric displacement field (coulomb/meter2)
B : magnetic field, also called the magnetic flux density (tesla, or volt-seconds/meter2)
A : surface area (meter2)
In linear, isotropic, nondispersive materials, D and B are related to E and H by:
D = ε E = ε0εrE and B = µ H = µ0µr
H (3.9)
where ε and µ are the medium permittivity and permeability, ε0 and µ0 are the per-
mittivity and permeability in free space, and εr and µr are the relative permittivity and
permeability.
Substituting (3.9) into (3.1) and (3.2) yields Maxwell’s curl equations in linear, isotropic,
nondispersive materials:∂ H
∂t= − 1
µ∇× E − 1
µσ∗ H (3.10)
∂ E
∂t=
1ε∇× H − 1
εσ E (3.11)
Writing out the vector components of the curl operators of (3.10), (3.11) yields the
following system of six coupled scalar equations under Cartesian coordinate:
∂Hx
∂t= − 1
µ
[∂Ez
∂y− ∂Ey
∂z
]− σ∗
µHx (3.12)
∂Hy
∂t= − 1
µ
[∂Ex
∂z− ∂Ez
∂x
]− σ∗
µHy (3.13)
∂Hz
∂t= − 1
µ
[∂Ey
∂x− ∂Ex
∂y
]− σ∗
µHz (3.14)
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 62
∂Ex
∂t=
1ε
[∂Hz
∂y− ∂Hy
∂z
]− σ
εEx (3.15)
∂Ey
∂t=
1ε
[∂Hx
∂z− ∂Hz
∂x
]− σ
εEy (3.16)
∂Ez
∂t=
1ε
[∂Hy
∂x− ∂Hx
∂y
]− σ
εEz (3.17)
The system of six coupled partial differential equations (3.12) - (3.17) forms the basis of
the FDTD numerical algorithm for electromagnetic wave interactions with general three-
dimensional objects.
Yee’s FDTD scheme discretizes Maxwell’s curl equations by approximating the time
and space first-order partial derivatives with centered differences using mesh and the
leapfrog scheme.
3.2.2 Yee’s Orthogonal Mesh
Yee’s algorithm centers its E and H components in a three-dimensional space so that
every E component is surrounded by four H components, and every H component is
surrounded by four E components. This provides a beautifully simple picture of three-
dimensional space being filled by an interlinked array of Faraday’s Law and Ampere’s
Law contours. It is possible to identify E components associated with displacement cur-
rent flux linking with H loops, as well as H components associated with magnetic flux
linking with E loops. This is shown in figure 3.1.
Utilizing Yee’s spatial gridding scheme, the spatial partial derivatives in equations
(3.12)-(3.17) can be approximated by central differential operators:
∂Ey
∂z|(i+ 1
2,j,k) ≈
Ey(i + 12 , j, k + 1
2 ) − Ey(i + 12 , j, k − 1
2)∆z
, etc. (3.18)
Consequently equation (3.12) becomes:
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 63
Figure 3.1: Yee’s spatial grid
∂Hx(i + 12 , j, k)
∂t+
1µ(i + 1
2 , j, k)· σ∗(i +
12, j, k)Hx(i +
12, j, k) = − 1
µ(i + 12 , j, k)
·[
Ez(i + 12 , j + 1
2 , k) − Ez(i + 12 , j − 1
2 , k)∆y
− Ey(i + 12 , j, k + 1
2 ) − Ey(i + 12 , j, k − 1
2 )∆z
]
(3.19)
3.2.3 Time Domain Discretization: the Leapfrog scheme and the Courant sta-
bility condition (CFL condition)
Yee’s algorithm also centers its E and H components in time. It is termed a leapfrog
arrangement (shown in figure 3.2). All of the E components in the modelled space are
computed and stored in memory using the previous E and the newly updated H data.
Then H is recomputed based on the previous H and the newly obtained E. This process
continues until time-stepping is concluded.
A central differential approximation is applied to equation (3.19):
∂Hx
∂t|(n∆t) ≈ H
(n∆t+ 12∆t)
x − H(n∆t− 1
2∆t)
x
∆t(3.20)
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 64
Figure 3.2: Leapfrog scheme - the temporal scheme of the FDTD method.
With approximation:
H(n∆t)x ≈ H
(n∆t+ 12∆t)
x + H(n∆t− 1
2∆t)
x
2(3.21)
equation (3.19) becomes a discretization equation (3.22) which can be solved easily
using computer program.
Hx(i +12, j, k)(n∆t+ 1
2∆t) =
1∆t − σ∗
2µ1
∆t + σ∗2µ
Hx(i +12, j, k)(n∆t− 1
2∆t)
− 1( 1∆t + σ∗
2µ) · µ∆y
[Ez(i +
12, j +
12, k)(n∆t) − Ez(i +
12, j − 1
2, k)(n∆t)
]
+1
( 1∆t + σ∗
2µ) · µ∆z
[Ey(i +
12, j, k +
12)(n∆t) − Ey(i +
12, j, k − 1
2)(n∆t)
]
(3.22)
Numerical stability of the Yee algorithm requires a bounding of the time-step (∆t) ac-
cording to the space increments (∆x, ∆y and ∆z). This is Courant-Friedrich-Levy (CFL)
stability condition, given in three dimensions by
∆t ≤ ∆tmax =1
c√
1∆x2 + 1
∆y2 + 1∆z2
(3.23)
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 65
In a cubic grid ( where ∆x = ∆y = ∆z = ∆), equation (3.23) can be expressed as
∆t ≤ ∆tmax =∆
c√
3(∆x = ∆y = ∆z = ∆) (3.24)
This upper bounding on ∆t enabled the successful application of FDTD methods to a
wide variety of electromagnetic wave modeling problems with moderate electrical size.
However, there are important potential applications of FDTD modelling which find the
Courant (CFL) stability bound too restrictive. For example, when simulating the fine-
scale geometric structures problems, the cell size ∆ need to be much less than the shortest
wavelength λmin. So for a fixed total time of simulation, the time step ∆t limited by CFL
can cause the total number of time-steps Nsim required to be very large:
Nsim =Tsim
∆t(3.25)
Hence, it makes the simulation computationally intensive and sometimes it is impos-
sible to achieve results in a reasonable simulation time.
Many attempts have been made to relax or even remove the stability constraint. Early
work involves applying alternating-direction-implicit (ADI) technique to the Yee’s grid
in order to formulate an implicit FDTD scheme [5]. In the first attempt of ADI-FDTD
in 1984 [5], the finite-difference operator was factored into three operators with each of
them being performed implicitly in respect to the three coordinate directions (namely x,
y, or z). However, this scheme was proved difficult to demonstrate numerical stability
at that time at that time [3]. It was from 1999, when a 2-D FDTD algorithm free of the
Courant stability conditions was proposed for a 2D-TE wave [22], the ADI method was
again applied. The resulting FDTD formulation was found to be unconditionally stable
[7, 22]. Consequently, the CFL constraint of the FDTD modeling is completely removed
and hence the selection of the time step is only dependent on the model accuracy required
[7].
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 66
3.3 Other Spatial Domain Discretization Schemes
Since the FDTD method is a grid based algorithm, mesh generation is very important.
A properly defined mesh will reduce numerical errors and increase computational effi-
ciency.
Orthogonal uniform meshes in the Cartesian coordinates are the most simple and
straightforward meshing scheme and they are most commonly used in FDTD modelling.
An orthogonal FDTD grid generally introduces the minimum numerical errors [113]. As
a result, even in a conformal mesh generation scheme, a boundary-orthogonal mesh is
preferred. However, the staircase approximation for curve structures often causes nu-
merical inaccuracy (or numerical dispersion) [114] and inefficiency.
A variety of mesh generation schemes have been developed, leading to many modi-
fied FDTD schemes. Beside the orthogonal staircase mesh, subgridding, nonorthogonal
meshes and hybrid meshes are all interesting candidates and have a wide range of appli-
cations, including problems where large structures with fine details and objects compris-
ing of curved or oblique surfaces.
3.3.1 Subgridding Mesh
In order for the numerical computation to yield accurate results, the spatial increment
∆x used in FDTD need to be much smaller (less than λ/10) than the structure of interest.
Consequently, simulating an electrically large domain with locally fine structures using
overall an fine mesh (small ∆x) and hence a small ∆t is computationally costly.
A subgridding mesh scheme is introduced to alleviate this problem. The basic idea
is to divide the computational volume into sub-regions and simulate them with variable
step sizes. A coarse grid is used in a large volume and fine meshes are applied only
around fine structures or discontinuities. To minimize the numerical reflections from an
abrupt transition from coarse meshes to fine ones, small scaling factors and a sequence
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 67
of subgrids can be applied when necessary [115].
The field updates can be explained by referring to figure 3.3. The fields inside the
coarse mesh sub-region and the fine mesh sub-region ( denoted in figure 3.3 by × and
black respectively), are calculated using the conventional FDTD equations (3.26) and
(3.27). The time increments in each sub-regions can be chosen according to the smallest
spatial increment or can be related to the spatial increments in each sub-region. On the
coarse-fine grid boundary, an interpolation is utilized to calculate the tangential electric
field (⊗) and the boundary layer magnetic fields in fine mesh (blue ). For the subgrid-
ding FDTD, various mesh and interpolation schemes can be found in [115–118].
The subgridding mesh scheme decreases the required computer memory and there-
fore expands the capability of the FDTD method. Meanwhile, it shows good numerical
stability [8].
Hn+ 1
2x (i, j, k) = H
n− 12
x (i, j, k)
− ∆t
µ
[En
z (i, j, k) − Enz (i, j − 1, k)
∆y− En
y (i, j, k) − Eny (i, j, k − 1)
∆z
]
Hn+ 1
2y (i, j, k) = H
n− 12
y (i, j, k)
− ∆t
µ
[En
x (i, j, k) − Enx (i, j, k − 1)
∆z− En
z (i, j, k) − Enz (i − 1, j, k)
∆x
]
Hn+ 1
2z (i, j, k) = H
n− 12
z (i, j, k)
− ∆t
µ
[En
y (i, j, k) − Eny (i − 1, j, k)
∆x− En
x (i, j, k) − Enx (i, j − 1, k)
∆y
](3.26)
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 68
En+1x (i, j, k) = En
x (i, j, k)
+∆t
ε
⎡⎣H
n+ 12
z (i, j + 1, k) − Hn+ 1
2z (i, j, k)
∆y− H
n+ 12
y (i, j, k + 1) − Hn+ 1
2y (i, j, k)
∆z
⎤⎦
En+1y (i, j, k) = En
y (i, j, k)
+∆t
ε
⎡⎣H
n+ 12
x (i, j, k + 1) − Hn+ 1
2x (i, j, k)
∆z− H
n+ 12
z (i + 1, j, k) − Hn+ 1
2z (i, j, k)
∆x
⎤⎦
En+1z (i, j, k) = En
z (i, j, k)
+∆t
ε
⎡⎣H
n+ 12
y (i + 1, j, k) − Hn+ 1
2y (i, j, k)
∆x− H
n+ 12
x (i, j + 1, k) − Hn+ 1
2x (i, j, k)
∆y
⎤⎦
(3.27)
Equations (3.28) and (3.29) show examples of how the field at the coarse-fine grid
boundary are updated using the neighbouring averaging scheme, subjected to the same
time interval calculated according to the fine mesh being used in all the sub-regions (see
figure 3.4).
Ef (2, 1, 1) =14Ec(1, 1) +
34Ec(2, 1) (3.28)
Hc(2, 2) =14Hf (2, 1, 2) +
14Hf (2, 2, 2) +
14Hf (2, 3, 2) +
14Hf (2, 4, 2) (3.29)
where Ef and Hf denote the fields in the fine mesh sub-region and Ec and Hc denote the
fields in the coarse mesh sub-region.
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 69
Figure 3.3: A cross section of a computational domain meshed according to the subgriddingalgorithm. Positions where the field quantities are calculated are shown. Since the spatialincrement in the fine mesh is only half that of the coarse grid, the time increment for the finemesh domain is equal to half of that in the coarse domain [8].
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 70
Figure 3.4: Enlarged view of the top-left corner of figure 3.3. In the coarse-fine grid boundary,the electric field in fine mesh (Ef (2, 1, 1)) is updated by electric field in coarse mesh region(Ec) using the neighbouring averaging equation (3.28); and the magnetic field in coarse mesh(Hc(2, 2)) is updated by magnetic field in the fine mesh region (Hf ) using the neighbouringaveraging equation (3.29).
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 71
3.3.2 Nonorthogonal Mesh
Many real world electromagnetic problems are characterized by geometries with oblique
angles or curved boundaries. Representing an oblique (or curved) surface using stair-
cased mesh usually requires very fine meshes which results in very small time steps in
the FDTD algorithm and consequently the computation becomes extensive.
As Maxwell’s equations are vector equations and can be implemented in any coordi-
nate system, they can be expressed in the nonorthogonal coordinate system as is shown in
[119]. In 1983, Holland introduced Maxwell’s equations in the nonorthogonal coordinate
system into the FDTD method and put forward a nonorthogonal FDTD (NFDTD) algo-
rithm, meshed with general nonorthogonal grids in a nonorthogonal coordinate system
[5]. In this mesh scheme, oblique surfaces or curved structures are meshed conformally
and more accurately with a coarser mesh. Since then, the NFDTD method has been re-
fined by many researchers including Yee [120], Lee [121], Mittra [122], Jurgens [123, 124],
Railton [55], Hao [18] and Douvalis [125] etc..
In nonorthogonal coordinate systems, an arbitrary vector can be expressed as a lin-
ear combination of two types of components according to two bases: the covariant and
contravariant components of this vector. In the FDTD modelling of an electromagnetic
(EM) problem, the covariant component relates physically to the flow of the vector along
the contour of an arbitrary surface while the contraviant component represents the flux
density of this vector passing through this surface.
Figure 3.5: A part of a three dimensional nonorthogonal mesh showing the covariant vectorswith the blue arrows and the contravariant vectors with the orange arrows.
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 72
However, compared with the conventional Cartesian FDTD method, the global curvi-
linear FDTD must store many additional metric tensors which are essential parameters in
the NFDTD scheme and are calculated from the spatial increment of each grid. While the
contravariant components are being updated in a Maxwellian way as in Yee’s scheme,
the covariant components have to be computed from the contravariant ones using two
additional projection equations applying the metric tensors.
As a result, the global curvilinear FDTD method is more computationally intensive
than the conventional Cartesian FDTD method. To alleviate this problem, a hybrid mesh
scheme, i.e. the Local Distorted NFDTD (LD-NFDTD) [18, 55] can be used. More details
of NFDTD method can be found in Chapter 4.
A mesh generating software GENGRID V2.2 is employed in this study. It is a free
software distributed by Computer Applied Fluid (CAF) Lab for educational purpose. It
is convenient in generating 2-D meshes, with four optional mesh stretching algorithms
provided.
3.3.3 Hybrid FDTD Meshes
Since each mesh generating scheme has its own advantages and drawbacks, it is quite in-
teresting to combine different mesh generation schemes to achieve efficient and accurate
FDTD grids for numerical simulations. An overview of example hybrid mesh generation
schemes is presented below.
• A Hybrid Conformal and Orthogonal Grid
This kind of mesh can be implemented by employing conformal cells at the curved
boundary within an underlying Cartesian coordinate system [126]. In other words, the
curvilinear meshes are used in the immediate vicinity of the curved boundary, while the
vast majority of the mesh away from the curved boundary can be rectangular/square.
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 73
Consequently, the additional computations from the NFDTD method are reduced. The
so-called ’Local-distorted nonorthogonal FDTD (LNFDTD)’ scheme also benefits from
greater accuracy and versatility than the conventional Yee’s Cartesian FDTD method but
with preserved efficiency [18].
• A Conformal Grid with the subgridding meshes
The subgridding in space domain can be applied combined with the conformal grid
leading to a subgridding NFDTD method. The computational efficiency is expected to
be improved in comparison with a subgridding scheme based on orthogonal mesh, or a
pure NFDTD scheme. Besides the subgridding in space, a time subgridding scheme can
also be employed in the NFDTD method and is reported to be helpful in reducing the
late time instability in the NFDTD method [126].
• A Conformal Grids with a triangular meshes
In [127], Schuhmann et al. observed that degenerated cells on the nonorthogonal
FDTD meshes are responsible for local field errors, leading to an irregular convergence
behaviour and also making the late time instability worse. To overcome this problem,
it is suggested that the NFDTD be combined with the triangular mesh fillings. Figure
3.6 demonstrates the staircase approximation, the nonorthogonal mesh, the degenerated
cell in the nonorthogonal mesh, and the proposed triangular fillings NFDTD respectively,
when meshing the cylindrical cavity. This scheme is regarded as a more flexible scheme
and may considerably improve the efficiency and accuracy of the NFDTD [127].
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 74
Figure 3.6: Meshes of the cross section of a cylindrical cavity. (a) the staircase approximation(b) nonorthogonal mesh (c) details of the degenerated cell in the nonorthogonal mesh whichis most responsible for the numerical error and late time instability; (d) the triangular fillingsNFDTD [127].
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 75
3.4 Boundary Conditions
In numerical modelling, many geometries of interest are defined in ’open’ regions where
the spatial domain of the computed field is unbounded in one or more coordinate di-
rections. Computers can only store and calculate a limited amount of data, therefore
an absorbing boundary condition (ABC) is often used to truncate the computational do-
main and to suppress spurious reflections of the outgoing numerical wave analogs to an
acceptable level.
Generally, there are two groups of ABCs: some derived from differential equations
and others employing absorbing materials. The most popular ABCs in the first group is
the one derived by Engquist and Majda [128] with the discretisation given by Mur[129].
These are based on approximating the outgoing wave equation by linear expressions
using a Taylor approximation. Material-based ABCs are realized by surrounding the
computational domain with a lossy material that dampens the outgoing fields. In this
group, the perfectly matched layer (PML) technique [130–135] which was put forward
by Berenger in 1994, demonstrated significantly better accuracy than most other ABCs so
it is widely used in the FDTD simulations.
An ideal EBG structure which has infinite periodicity does not exist in the real world.
Any EBG structure used in the real world is with boundaries. However, it is always
interesting and beneficial to study an infinite EBGs before the application of its finite-
sized dual. Periodic boundary condition (PBC) is the tool which enable the computation
of an infinite EBG through a limited computational domain. So EBGs which are infinite
in one-dimension, two-dimension and three-dimension can be modelled efficiently by
studying a few unit cells (or one unit cell) and applying proper PBCs.
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 76
3.4.1 Mur’s Absorbing Boundary Conditions (ABCs)
Engquist and Majda derived a theory of one-way wave equations which permits wave
propagation only in certain directions. For example, consider the two-dimensional wave
equations in Cartesian coordinates:
∂2U
∂x2+
∂2U
∂y2− 1
c2
∂2U
∂t2= 0 (3.30)
where U is a scalar field component and c is the wave phase velocity. We can define the
partial differential operator as:
L =∂2
∂x2+
∂2
∂y2− 1
c2
∂2
∂t2= D2
x + D2y −
1c2
D2t (3.31)
The wave equation is then written as
LU = L−L+U = 0 (3.32)
with the wave operator L been factored by L− and L+:
L− ≡ Dx − Dt
c
√1 − S2 and L+ ≡ Dx +
Dt
c
√1 − S2 (3.33)
with
S =Dy
Dt/c(3.34)
Solution of equation (3.35)
L−U = 0 (3.35)
is a solution of the wave function (3.30) however the wave only propagates towards the
−x direction. So, if tangential field components at the boundary of x = 1 satisfy equation
(3.35), this wave is not bouncing back into the computational domain.
As the space increment ∆ being much smaller than the smallest operating wave-
length, the variation of the the field components along grids should be small. So when
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 77
S is small, Taylor series can be used to approximate the square-root function in (3.33) by
two-term expansion: √1 − S2 ∼= 1 − 1
2S2 (3.36)
Substituting equation (3.36) into equation (3.33) yields:
L− = Dx − Dt
c
[1 − 1
2
(cDy
Dt
)2]
= Dx − Dt
c+
cD2y
2Dt(3.37)
Then substituting (3.37) into equation into (3.35), multiplying by Dt we get a second-
order accurate ABC at the x = 0 boundary.
DtL−U =
∂2U
∂x∂t− 1
c
∂2U
∂t2+
c
2∂2U
∂y2= 0 (3.38)
Mur used a simple central-difference scheme to interpret (3.38) in Yee’s space (with
spatial increment ∆x,∆y) and time (with time step ∆t) domain. For example, the mixed
x and t derivative in the second order ABC is written as:
∂2U |n1/2,j
∂x∂t=
12∆t
[(U |n+1
1,j − U |n+10,j
∆x
)−(
U |n−11,j − U |n−1
0,j
∆x
)](3.39)
In this way, the tangential field under discretization at the boundary (e.g. U |n+10,j ) is calcu-
lated as follows ( suppose ∆x = ∆y = ∆):
U |n+10,j = −U |n+1
1,j +c∆t − ∆c∆t + ∆
(U |n+1
1,j + U |n−10,j
)+
2∆c∆t + ∆
(U |n1,j + U |n0,j
)+
(c∆t)2
2∆(c∆t + ∆)(U |n0,j+1 − 2U |n0,j + U |n0,j−1
)+
(c∆t)2
2∆(c∆t + ∆)(U |n1,j+1 − 2U |n1,j + U |n1,j−1
)(3.40)
Remove the y-directive term, the first-order Mur’s ABC at x = 0 boundary is ob-
tained:
U |n+10,j = −U |n+1
1,j +c∆t − ∆c∆t + ∆
(U |n+1
1,j + U |n−10,j
)+
2∆c∆t + ∆
(U |n1,j + U |n0,j
)(3.41)
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 78
3.4.2 Perfect Matched Layers (PML)
In the perfect matched layer (PML)[130, 131] medium, each component of the electro-
magnetic field is split into two parts. In Cartesian coordinates, the six components yield
and Maxwell’s equations are replaced by 12 equations,
ε∂Exy
∂t+ σyExy =
∂Hzx + Hzy
∂y(3.42)
ε∂Exz
∂t+ σzExz = −∂Hyz + Hyx
∂z(3.43)
ε∂Eyz
∂t+ σzEyz =
∂Hxy + Hxz
∂z(3.44)
ε∂Eyx
∂t+ σzEyx = −∂Hzx + Hzy
∂x(3.45)
ε∂Ezx
∂t+ σzEzx =
∂Hyz + Hyx
∂x(3.46)
ε∂Ezy
∂t+ σzEzy = −∂Hxy + Hxz
∂y(3.47)
µ∂Hxy
∂t+ σ∗
xHxy = −∂Ezx + Ezy
∂y(3.48)
µ∂Hxz
∂t+ σ∗
yHxz =∂Eyz + Eyx
∂z(3.49)
µ∂Hyz
∂t+ σ∗
xHyz = −∂Exy + Exz
∂z(3.50)
µ∂Hyx
∂t+ σ∗
yHyx =∂Ezx + Ezy
∂x(3.51)
µ∂Hzx
∂t+ σ∗
xHzx = −∂Eyz + Eyx
∂x(3.52)
µ∂Hzy
∂t+ σ∗
yHzy =∂Exy + Exz
∂y(3.53)
where the parameters (σx, σy, σz , σ∗x, σ∗
y , σ∗z ) are homogeneous electric and magnetic con-
ductivities. Applying the central discretization to the temporal and spatial partial dif-
ferential operator yields FDTD equations with PML absorbing boundary. For example,
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 79
equation (3.42) becomes
En+ 1
2xy (i, j, k) =
1∆t − σy
2ε1
∆t + σy
2ε
En− 1
2xy (i, j, k) +
1( 1∆t + σy
2ε ) · ε∆y
· [Hnzx(i, j + 1, k) − Hn
zx(i, j, k) + Hnzy(i, j + 1, k) − Hn
zy(i, j, k)](3.54)
It is proven that for any propagating plane wave at an interface normal to a (a, b, c =
x, y, z) lying between PML media of same ε and µ, if the transverse conductivities σb,σ∗b ,σc,σ∗
c
are equal and all couples of conductivities (σx, σ∗x),(σy, σ
∗y),(σz, σ
∗z ) satisfy the matching
impedance condition (σ/ε = σ∗/µ), the wave will be transmitted into and in between the
PML layers with no reflection.
This approach is based on the splitting of the field components into two sub-components.
Sacks et al. [132], Gedney [133] etc. were able to formulate the PML based on a Maxwellian
formulation which removed the need to split the field. Veihl and Mittra presented a sim-
ilar work in a different way [134]. The application of the unsplit field PML to the FDTD
method is more computationally efficient, and it can be extended to nonorthogonal and
unstructured grid techniques.
3.4.3 Periodic Boundary Condition (PBC)
As is mentioned previously, in an infinite EBG structure, the wave or field is in Bloch’s
state and can be studied by the unit cell approach, in which only elements in one unit cell
are modelled and the fields in the adjacent unit cells are expressed explicitly using the
periodic boundary condition[76] (PBC):
F (r + la) = eik·la F (r) (3.55)
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 80
where a is the lattice constant, l is any vector with the same dimensions with a and all
entries integers, and F can be the covariant EM field components or the contravariant
EM fluxes.
Equation (3.55) is an important boundary condition in EBG modelling because it en-
ables an efficient numerical approach to analyze the infinite EBG structures. When the
EM field components of the cells on the boundary of the FDTD computational domain
are updated, the field components outside the computational domain will be referred to,
which, by utilizing PBCs, can be expressed using the field components inside the com-
putational domain and introducing a phase shift calculated from the dimension of the
unit cell. For example, equation (3.56) shows how Hx at boundary k = 1 is updated us-
ing PBCs when variable Ey(i, j, 0) is not available in the FDTD domain. Equation (3.57)
shows the updating equation for Ex at boundary k = maxz (where maxz denotes the
maximum number in z direction in the FDTD domain).
Hn+ 1
2x (i, j, 1) = H
n− 12
x (i, j, 1)
−∆t
µ
[En
z (i, j, 1) − Enz (i, j − 1, 1)
∆y− En
y (i, j, 1) − Eny (i, j,maxz) · e−ik·maxz∆z
∆z
]
(3.56)
En+ 1
2x (i, j,maxz) = E
n− 12
x (i, j,maxz)
+∆t
ε
[Hn
z (i, j + 1,maxz) − Hnz (i, j,maxz)
∆y− Hn
y (i, j, 1) · eik·maxz∆z − Hny (i, j,maxz)
∆z
]
(3.57)
The other equations for the field components at all the boundaries can be updated in
this way. Figure 3.7 (a) and (b) illustrate the interpolations of PBCs in EBGs with square
and triangular lattices respectively.
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 81
(a)
(b)
Figure 3.7: Periodic boundary conditions when calculating the infinite EBGs. The inclusionof the EBGs (marked by the solid black line) can be of any shape. The FDTD computationaldomain is limited to one unit cell/ super cell, marked by the green color. To calculate the fieldof the boundary layer of the computational domain (Layer 1 or Layer maxv in the graph),fields at the adjacent unit cell/ super cells are needed but they are outside the computationaldomain (marked with the yellow layer). However, they can be expressed using the fieldvalue within the domain (Layer maxv or Layer 1) applying equation (3.55). (a) PBCs in EBGswith rectangular lattice. The interpolations between cells are marked with the orange andthe pink lines. (b) PBCs in EBGs with triangular lattice. The interpolations between cells aremarked with the orange lines.
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 82
3.5 Band Gap Calculation
Through the FDTD modelling of the bandgap structures, some EBG characteristics can
be obtained in an intuitive way. For example, the pass-band/stop-band behaviour and
the transmission/reflection coefficients can be examined readily by the time-domain field
response. However, to study the dispersion relation (bandgap characteristics) of an EBG
structure, certain procedures involving post processing are necessary (see figure 3.8) and
will be presented in this section.
3.5.1 Source Excitation
A Modulated Gaussian Pulse (also termed as the Gabor pulse) with the following form
is used in the simulation of EBG structures:
S(t) = e−t2
2σ2 cos(2πξt + φ) (3.58)
where σ controls the (effective) time width of the pulse and consequently controls
the bandwidth of the source [136]. ξ and φ are the frequency and phase of the single
frequency wave that modulates the Gaussian pulse. Figure 3.9 shows a typical modulated
Gaussian waveform with σ = 8 × 10−6 s (= 40∆t; ∆t: sampling time), ξ = 100kHz, and
φ = 0 rad.
A modulated Gaussian pulse is chosen because it is efficient in time-frequency reso-
lution. Compared with a truncated pure sine pulse, the pulse energy of the modulated
Gaussian pulse is more concentrated near the center time of the pulse and the center fre-
quency. By controlling the time width σ, a desired form of a modulated Gaussian pulse
for a certain application can be defined.
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 83
Figure 3.8: The FDTD procedure in modelling EBG structures.
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 84
(a)
(b)
Figure 3.9: The modulated Gaussian pulse. (a)The shape of the modulated Gaussian pulsein the time domain and (b) the magnitude of its Fourier transform [136].
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 85
3.5.2 Dispersion Diagram Calculation
The dispersion diagram is very useful in the study of the bandgap characterization of an
infinite EBGs. As is introduced in last chapter, the dispersion diagram plots the possible
modes against the wave vector in the irreducible Brillouin zone. With proper periodic
boundary conditions, the infinite EBGs can be modelled by only one cell, or only a group
of cells if there are defects in the EBGs. The cell or the group of cells will be referred to in
the following paragraphs as the unit cell or the super cell respectively [61].
• The Unit Cell Approach
After a mesh is set up for the unit cell, one can randomly choose a few points in the
mesh as source points [76] and several points as probe (or monitored) points. The probe
points should be evenly located and dense enough to capture all the possible modes that
will be generated. The k vector from the irreducible Brillouin zone is used to set up the
periodic boundary conditions.
A modulated Gaussian pulse is applied at the source points to excite all the possible
electromagnetic (EM) modes of the EBG over a wide range of frequencies. As the FDTD
time evolution proceeds, only the true transmission modes remain in the computational
domain, and the pseudo transmission modes will eventually vanish [79]. After the tem-
poral responses of the probe points are recorded at every time step for a proper period
of time, the temporal signatures are Fourier transformed to obtain frequency spectra on
which peaks at certain frequency values can be found. These peaks indicate the exis-
tence of the supported transmission modes (eigenmodes of the EBGs) corresponding to
the wave vector k. Plotting these frequency values against the wave vector k gives the
dispersion diagram of the EBGs.
• The Super Cell Approach
The super cell approach works almost the same way as the unit cell approach does,
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 86
only the modelling domain consists of the defected cell(s) normally surrounded by reg-
ular unit cells. Periodic boundary conditions (PBCs) are also used to terminate the mod-
elling domain if the structure is assumed to be infinite. In the direction in which the
defects appears periodically, the PBCs are placed one period (of the defects) away from
each other. In the direction where the defects do not show periodicity, a number of unit
cells will be necessary in between the PBCs. Since the use of PBCs will represent the
defect periodically where it does not actually appear, the number of EBG unit cell layers
should be large enough to isolate the EM modes from the spurious defects in the neigh-
bouring super cell. On the other hand this unit cell layer should be minimized as much
as possible in order to maximize the efficiency of the computation. This number is often
determined experientially and is normally more than 10 [79]. Figure 3.10 shows the su-
per cell used by Chutinan et al. when they model the waveguide created by filling up one
column of the air holes in the EBGs [61].
Figure 3.10: The super cell of the waveguide created by filling up one column of the air holesin the EBGs [61]. In the y direction, the defects are periodic with period of one unit cell, soone unit cell is used in between the PBCs. In x direction, five unit cells are used between thePBCs to isolate the modes from the neighbouring spurious defects.
3.5.3 Transmission and Reflection Coefficient Calculation
If the bandgap structure is not infinite in its periodic direction, then transmission coeffi-
cient is helpful in order to find the band gap of the structure of interest.
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 87
Take the two-dimensional EBG with 4 arrays of cylindrical rods in x-direction and
infinitely loaded rods in y-direction in free space for example. Since the rods are infinite
in y-direction, the periodic boundary condition (PBC) can still be used to terminate the
computational domain in this direction. In x-direction, in which there exist 4 arrays of
rods, absorbing boundary condition (ABCs) can terminate the computational domain at
a proper distance away from the scattering structure. In this way of combining PBCs
with ABCs, the computational efficiency of the finite EBGs modelling can be increased.
The boundary condition setup is shown in figure 3.11.
Figure 3.11: Numerical Model for a two-dimensional EBG structure of semi-finite size. Thereare 4 arrays of cylindrical rods in x-direction and infinitely loaded rods in y-direction. So thecomputation domain is terminated by the PBCs in y-direction and by ABCs in x-direction. Aplane wave source in form of Modulated Gaussian Pulse is defined at one side of the EBGs.The responses at the other side of the EBGs are collected as Probe set 1 for calculating thetransmission coefficient (S21). The responses at the same side of the EBGs are collected asProbe set 2 for calculating the reflection coefficient (S11).
A plane wave travelling in the x-direction in form of a Modulated Gaussian pulse
is excited from a line source at one end of the structure. Temporal signatures of two
lines of probes at both sides of the structures (as shown in figure 3.11) are recorded and
analyzed to calculate the transmission and reflection coefficients respectively. For probe
sets 1, the Fourier Transformation can be applied directly to the time domain signal.
The averaged frequency spectra along the probe line shows the transmission coefficient
against frequency and bandgap can be found. To calculate the reflection coefficient, the
process is similar only the signal is recorded after the first pulse passed.
If the whole structure is finite in size, then the whole computational domain should
be terminated with ABCs in all directions. A modulated Gaussian pulse is excited in one
side of the EBGs (point source or line source) and the probes are defined at the other side.
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 88
Another simulation is required with the EBG structure replaced by free space. Then the
band gap can be found with the transmission coefficient from the EBGs calibrated by that
from the free space model (figure 3.12).
Figure 3.12: Numerical Model for an EBG structure of finite size. The computational domainis terminated by the ABCs in all the directions. A Modulated Gaussian Pulse is excited at oneside of the EBGs. The responses at Probe set 1 and Probe set 2 with the existence of the EBGare calibrated by those without EBG in the calculation of the transmission coefficient (S21)and the reflection coefficient (S11).
Chapter 3 A Brief Introduction to the Finite-Difference Time-Domain Method for Modelling the EBG Structures 89
3.6 Summary
The Finite-Difference Time-Domain (FDTD) method is widely used because of its sim-
plicity for numerical implementation. It is a flexible means of directly solving Maxwell’s
time-dependent curl equations or their equivalent integral equations using the finite dif-
ference approximations as well. It can be used to solve various types of electromagnetic
problems, including the anisotropic and nonlinear problems. This chapter briefly re-
viewed the fundamentals about the FDTD method, including Yee’s spatial and temporal
grid, the updating formulation and two important boundary conditions - the absorbing
boundary conditions (ABCs) and the periodic boundary conditions (PBCs). The comput-
ing techniques specifically in calculating the EBG related parameters are also presented,
including the dispersion diagram calculation, and the transmission and reflection coeffi-
cient calculation.
Developments on the FDTD method towards a more accurate and more computa-
tionally efficient method never stop evolving. The next chapter will review two major
enhanced schemes over the Yee’s FDTD approach. A novel FDTD approach based on
these enhanced schemes is proposed.
Chapter 4
The Development of the
Alternating-Direction Implicit
Nonorthogonal FDTD Method
4.1 Introduction
Despite the simplicity and flexibility of the original Yee’s FDTD algorithm, the FDTD
applications require large amount of memory and central processing unit (CPU) time to
obtain accurate solutions when solving electrically large structure problems. Theoretical
studies on the FDTD show that the intensive memory and CPU time requirement mainly
come from the following two modelling constrains[3, 137]. Firstly, the spatial increment
step must be small enough in comparison with the wavelength (usually 10 ∼ 20 steps
per smallest wavelength) in order to obtain accurate field component values. Secondly,
the time step must be small enough to meet the Courant-Friedrich-Levy (CFL) stability
condition. If the time step is beyond the CFL bound, the FDTD scheme will become
numerically unstable leading to a spurious increase of the field values without limit as a
FDTD solution marches [138].
90
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 91
There are two major extensions of the FDTD method addressing these two aforemen-
tioned constrains - the Nonorthogonal FDTD (NFDTD) method [5, 121] which improves
modelling accuracy while decreasing the requirement of the spatial increment, and the
FDTD algorithm based on the Alternating Direction Implicit method (ADI-FDTD) which
releases the CFL restriction on the FDTD time step[138].
Based on these two methods, a novel Alternating-Direction Implicit Nonorthogonal
Finite-Difference Time-Domain method (ADI-NFDTD) is developed. In the proposed
method, the curved structures are modelled based on the NFDTD algorithm. However,
by the introduction of the implicit method, the ADI-NFDTD method has demonstrated
an improvement in terms of the late time instability which is inherent in the NFDTD
method. This chapter will firstly review the NFDTD and the ADI-FDTD methods. Then
the ADI-NFDTD method is introduced and validated using numerical simulations.
4.2 A Brief Introduction to the Nonorthogonal Finite-Difference
Time-Domain Method
In 1983, Holland developed a nonorthogonal FDTD (NFDTD) algorithm and opened up
the possibilities of a more general, efficient and accurate numerical method [5]. In this
method, the FDTD technique is no longer restricted to an orthogonal cartesian grid. In-
stead, a generalized curvilinear coordinate system is used. As a direct consequence, an
arbitrary structure with curved boundary or oblique surface can be meshed conformally
and be modelled accurately, without employing the staircase approximation as is the
case in the Yee’s algorithm. Since the NFDTD method was proposed, it has been success-
fully applied to analyze optical dielectric waveguide, dielectric-loaded resonant cavity,
microstrip discontinuities and periodic structures at oblique incidence, etc.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 92
In this section, the basic formulations of the NFDTD [5, 121] and its recent develop-
ment [139] will be reviewed. The numerical instability problem is also discussed.
4.2.1 The Curvilinear Coordinate Systems
Since Maxwell’s equations can be implemented in any coordinate system, the nonorthog-
onal coordinate system is introduced into the conventional FDTD method and a global
distorted grid conformal to the curved boundary is generated [119]. Therefore geometri-
cal characteristics involved in the spatial domain are clearly defined without the staircase
approximation. This section reviews the basics of the nonorthogonal coordinate system.
• Base vectors
(a) (b)
Figure 4.1: The definition of basic vectors (a) covariant vector; (b) contravariant vector.
There are two types of base vectors in the nonorthogonal coordinate system: covari-
ant and contravariant bases [113, 119]. Consider a coordinate line along which only the
coordinate ξ varies (see figure 4.1(a)), a tangential vector to the coordinate line is given
by:
limdξ→0
r(ξ + dξ) − r(ξ)dξ
= rξ (4.1)
The covariant bases in the nonorthogonal coordinates are tangential to the three co-
ordinate lines along which only ξi(i = 1, 2, 3) varies, designated as:
ai = rξi (4.2)
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 93
A normal vector to a coordinate surface on which the coordinate ξ is constant is given
by (see figure 4.1(b)). These normal vectors to the three coordinate surfaces are the three
contravariant base vectors of the curvilinear coordinate system, designated as:
ai = ∇ξi (4.3)
Figure 4.2 illustrates two types of base vectors for a nonorthogonal FDTD cell with
six sides.
Figure 4.2: The covariant basic vectors (a1, a2 and a3), and the contravariant basic vectors(a1, a2 and a3) in a nonorthogonal FDTD cell.
Then, an arbitrary vector can be expressed by:
E =3∑
i=1
eiai =3∑
i=1
eiai (4.4)
And the relationship holds:
ai · aj = δij (4.5)
where δ is the Kronecker delta:
δij =
⎧⎨⎩ 1 for i = j
0 for i = j, (4.6)
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 94
• Metric tensor
The metric tensor plays an important role and is also sufficient to characterize com-
pletely the geometrical properties of the object. Using the notation of Stratton [119], in the
curvilinear space a vector can be represented by its covariant and contravariant tensors,
as defined in equation (4.7)-(4.8).
¯gij =
⎡⎢⎢⎢⎣
g11 g12 g13
g21 g22 g23
g31 g32 g33
⎤⎥⎥⎥⎦
¯gij =
⎡⎢⎢⎢⎣
g11 g12 g13
g21 g22 g23
g31 g32 g33
⎤⎥⎥⎥⎦
where
gij = ai · aj (4.7)
andgij = ai · aj (4.8)
From equations (4.4) - (4.6) and equations (4.7), (4.8), the covariant and contravariant field
components are related to each other by the use of the matric tensors:
ei =3∑
j=1
gijej (4.9)
ei =3∑
j=1
gijej (4.10)
Denote g as :
√g = a1 · a2 × a3 . (4.11)
The reciprocal basis related to the unitary basis as
ai =aj × ak√
g( i,j,k in ascending order). (4.12)
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 95
• Curl in the Nonorthogonal Coordinates
The operation of curl in the nonorthogonal coordinates can be written as [113]:
∇× A =1√g
3∑i=1
[(∂Ak
∂uj
)ai −
(∂Ak
∂ui
)aj
]
( i,j,k in ascending order. ) (4.13)
If a contravariant base vector am(m = 1, 2, 3) is used to perform a scalar dot product
of equation (4.12), then combining with equation (4.5), equation (4.12) becomes:
∇× A · am =1√g
3∑i=1
[(∂Ak
∂uj
)(ai · am) −
(∂Ak
∂ui
)(aj · am)
]
=1√g
3∑i=1
[(∂Ak
∂uj
)δim −
(∂Ak
∂ui
)δjm
]
( i,j,k in ascending order and m = 1, 2, 3.) (4.14)
Taking m = 1 for example:
∇× A · a1 =1√g
3∑i=1
[(∂Ak
∂uj
)(ai · a1
)− (∂Ak
∂ui
)(aj · a1
)]
=1√g
3∑i=1
[(∂Ak
∂uj
)δi1 −
(∂Ak
∂ui
)δj1
]
=1√g
[∂A3
∂u2− ∂A2
∂u3
](4.15)
Equation (4.14) is the basis of Holland’s differential equations. Lee reformulated Hol-
land’s scheme and presented a more efficient time marching scheme in the nonorthogonal
FDTD algorithm in 1992 [121]. Lee’s scheme dominates ever since, until recently when
an alternative scheme was developed by Douvalis [139], which is also presented in the
following.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 96
4.2.2 The Conventional Nonorthogonal FDTD
Based on the aforementioned basic concepts of nonorthogonal coordinates, Maxwell’s
equations can go through discretization in a nonorthogonal coordinate system. In a
source free and loss free medium, Maxwell’s curl equations are expressed as:
−µ∂ H
∂t= ∇× E (4.16)
ε∂ E
∂t= ∇× H (4.17)
Following equation (4.14), Maxwell’s curl equation can be written as:
−µ∂h1
∂t=
1√g
(∂e3
∂u2− ∂e2
∂u3
)
−µ∂h2
∂t=
1√g
(∂e1
∂u3− ∂e3
∂u1
)
−µ∂h3
∂t=
1√g
(∂e2
∂u1− ∂e1
∂u2
)
ε∂e1
∂t=
1√g
(∂h3
∂u2− ∂h2
∂u3
)
ε∂e2
∂t=
1√g
(∂h1
∂u3− ∂h3
∂u1
)
ε∂e3
∂t=
1√g
(∂h2
∂u1− ∂h1
∂u2
)(4.18)
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 97
In a similar procedure as in the Cartesian FDTD scheme, equations (4.18) are dis-
cretized using the central-difference approximations in both of the time and space do-
main. So equations (4.18) become:
h1(n+ 12)(i +
12, j, k) = h1(n− 1
2)(i +
12, j, k)
− ∆t
µ(i + 12 , j, k)
√g(i + 1
2 , j, k)·(
e(n)3 (i +
12, j +
12, k) − e
(n)3 (i +
12, j − 1
2, k))
+∆t
µ(i + 12 , j, k)
√g(i + 1
2 , j, k)·(
e(n)2 (i +
12, j, k +
12) − e
(n)2 (i +
12, j, k − 1
2))
h2(n+ 12)(i, j +
12, k) = h2(n− 1
2)(i, j +
12, k)
− ∆t
µ(i, j + 12 , k)
√g(i, j + 1
2 , k)·(
e(n)1 (i, j +
12, k +
12) − e
(n)1 (i, j +
12, k − 1
2))
+∆t
µ(i, j + 12 , k)
√g(i, j + 1
2 , k)·(
e(n)3 (i +
12, j +
12, k) − e
(n)3 (i − 1
2, j +
12, k))
h3(n+ 12)(i, j, k +
12) = h3(n− 1
2)(i, j, k +
12)
− ∆t
µ(i, j, k + 12)√
g(i, j, k + 12 )
·(
e(n)2 (i +
12, j, k +
12) − e
(n)2 (i − 1
2, j, k +
12))
+∆t
µ(i, j, k + 12)√
g(i, j, k + 12 )
·(
e(n)1 (i, j +
12, k +
12) − e
(n)1 (i, j − 1
2, k +
12))
e1(n+1)(i, j +12, k +
12) = e1(n)(i, j +
12, k +
12)
+∆t
ε(i, j + 12 , k + 1
2)√
g(i, j + 12 , k + 1
2 )·(
h(n+ 1
2)
3 (i, j + 1, k +12) − h
(n+ 12)
3 (i, j, k +12))
− ∆t
ε(i, j + 12 , k + 1
2)√
g(i, j + 12 , k + 1
2 )·(
h(n+ 1
2)
2 (i, j +12, k + 1) − h
(n+ 12)
2 (i, j +12, k))
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 98
e2(n+1)(i +12, j, k +
12) = e2(n)(i +
12, j, k +
12)
+∆t
ε(i + 12 , j, k + 1
2 )√
g(i + 12 , j, k + 1
2 )·(
h(n+ 1
2)
1 (i +12, j, k + 1) − h
(n+ 12)
1 (i +12, j, k)
)
− ∆t
ε(i + 12 , j, k + 1
2 )√
g(i + 12 , j, k + 1
2 )·(
h(n+ 1
2)
3 (i + 1, j, k +12) − h
(n+ 12)
3 (i, j, k +12))
e3(n+1)(i +12, j +
12, k) = e3(n)(i +
12, j +
12, k)
+∆t
ε(i + 12 , j + 1
2 , k)√
g(i + 12 , j + 1
2 , k)·(
h(n+ 1
2)
2 (i + 1, j +12, k) − h
(n+ 12)
2 (i, j +12, k))
− ∆t
ε(i + 12 , j + 1
2 , k)√
g(i + 12 , j + 1
2 , k)·(
h(n+ 1
2)
1 (i +12, j + 1, k) − h
(n+ 12)
1 (i +12, j, k)
)
(4.19)
Furthermore, in order to fulfill an explicit time-marching nonorthogonal FDTD scheme,
the covariant field components (ei/hi) must be calculated using its dual field (ei/hi). So,
additional projection equations (4.10) are needed. As the covariant field values should
be in the same space position with the contravariant ones, an interpolation scheme (i.e. a
neighbouring averaging projection scheme) is introduced as follows:
Since in the orthogonal FDTD method the grids are uniform and orthogonal, with
spatial increment ∆x and ∆y in x and y direction, equations (4.86) and (4.87) are obtained.
g12(i, j) = 0 (4.86)
g11(i, j) = ∆x2
g22(i, j) = ∆y2
g(i, j) = ∆x2 · ∆y2 (4.87)
Substituting equations (4.85) and (4.86) into (4.82), yields equation (4.88).
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 125
−√
g22(i − 1, j) · dt2
4µ(i − 1, j) ·√g(i, j)√
g(i − 1, j)·√
g22(i − 1, j) · D2n+ 12 (i − 1, j)
ε2(i − 1, j)
+
[ε2(i, j)√g22(i, j)
+
√g22(i, j) · dt2
4µ(i − 1, j) ·√g(i, j)√
g(i − 1, j)+
√g22(i, j) · dt2
4µ(i, j) ·√g(i, j)√
g(i, j)
]
·√
g22(i, j) · D2n+12 (i, j)
ε2(i, j)
−√
g22(i, j) · dt2
4µ(i, j) · g(i, j)·√
g22(i + 1, j) · D2n+ 12 (i + 1, j)
ε2(i + 1, j)
=ε2(i, j)√g22(i, j)
·√
g22(i, j) · D2n(i − 1, j)
ε2(i, j)
− dt
2√
g11(i, j) ·√
g22(i, j)· [Hn
3 (i, j) − Hn3 (i − 1, j)]
− dt2
4√
g11(i, j) ·√
g22(i, j)· [ 1
µ(i, j) ·√g22(i, j)· En
1 (i, j + 1) − En1 (i, j)√
g11(i, j)
− 1µ(i − 1, j) ·√g22(i − 1, j)
· En1 (i − 1, j + 1) − En
1 (i − 1, j)√g11(i − 1, j)
] (4.88)
Then by substituting (4.84) and (4.87) into equation (4.88) to change the variants to or-
thogonal FDTD variants and by multiplying −4∆x2·∆y·µ(i,j)
dt2on both sides of the resulting
equation, equation (4.89) is derived, which is ADI-FDTD formula under uniform orthog-
onal grid.
− µ(i, j)µ(i − 1, j)
· En+ 12
y⊥ (i − 1, j) − [4∆x2εy(i, j)µ(i, j)
dt2+
µ(i, j)µ(i − 1, j)
+ 1] · En+ 12
y⊥ (i, j)
+En+ 1
2y⊥ (i + 1, j)
=4∆x2εy(i, j)µ(i, j)
dt2· En
y⊥(i, j) − 2∆x · µ(i, j)dt
· [Hnz⊥(i, j) − Hn
z⊥(i − 1, j)]
+∆x
∆y· [En
x⊥(i, j + 1) − Enx⊥(i, j)] +
∆x
∆y· µ(i, j)µ(i − 1, j)
· [Enx⊥(i − 1, j + 1) − En
x⊥(i − 1, j)]
(4.89)
Equation (4.89) is the general equation for the orthogonal ADI-FDTD scheme. It can
be applied in homogeneous or inhomogeneous, isotropic or anisotropic media.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 126
4.4.3 Periodic Boundary Condition Incorporated in the ADI-NFDTD Method
The periodic boundary condition (PBC) for numerically modelling the infinite EBGs can
be easily introduced into the proposed ADI-NFDTD.
The PBCs in the explicit formulae in the ADI-NFDTD (equations (4.73), (4.75), (4.77)
and (4.78)) can be realized in the same way as in a conventional FDTD (or NFDTD).
The formulation of PBCs in the matrix calculation (equations (4.82) and (4.83) which are
derived from the implicit equations (4.74) and (4.76)) is presented in the following. x, y, z
are used to denote the three directions (dimensions) in stead of 1, 2, 3 in the following
part of the thesis. They do not indicate that the variables are under Cartesian coordinate.
Assume an infinite EBG modelling with elements aligned in square lattice. Assume
the unit cell consists of xmax cells in x-direction and ymax cells in y-direction respectively,
so that the computational domain is (1 ≤ i ≤ xmax, and 1 ≤ j ≤ ymax). Recall the
ADI-NFDTD implicit equation for the calculation of Dy for example:
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 127
− gyy(i − 1, j) · dt2
4µ(i − 1, j) · εy(i − 1, j) ·√g(i, j)√
g(i − 1, j)· Dyn+ 1
2 (i − 1, j)
+
[1 +
gyy(i, j) · dt2
4µ(i − 1, j) · εy(i, j) ·√
g(i, j)√
g(i − 1, j)+
gyy(i, j) · dt2
4µ(i, j) · εy(i, j) · g(i, j)
]
·Dyn+ 12 (i, j) − gyy(i + 1, j) · dt2
4µ(i, j) · εy(i + 1, j) · g(i, j)· Dyn+ 1
2 (i + 1, j)
= Dyn(i,j) − dt
2√
g(i, j)[Hn
z (i, j) − Hnz (i − 1, j)]
− dt2
4√
g(i, j)·[
Enx (i, j + 1) − En
x (i, j)µ(i, j)
√g(i, j)
− Enx (i − 1, j + 1) − En
x (i − 1, j)µ(i − 1, j)
√g(i − 1, j)
]
+dt2
16µ(i, j)g(i, j)
·⎡⎣gxy(i + 1, j)
εy(i + 1, j)
a=i+1;b=j+1∑a=i;b=j
Dxn+12 (a, b) − gxy(i, j)
εy(i, j)
a=i;b=j+1∑a=i−1;b=j
Dxn+12 (a, b)
⎤⎦
− dt2
16µ(i − 1, j)√
g(i, j)√
g(i − 1, j)
·⎡⎣gxy(i, j)
εy(i, j)
a=i;b=j+1∑a=i−1;b=j
Dxn+ 12 (a, b) − gxy(i − 1, j)
εy(i − 1, j)
a=i−1;b=j+1∑a=i−2;b=j
Dxn+12 (a, b)
⎤⎦ (4.90)
When updating fields in any cell (i, j) (1 < i < xmax and 1 < j < ymax) in the domain
of the unit cell except for the boundary (referred to as the ’main area’ in the following text
for short), the entries in the coefficient matrix W1(i, j), W2(i, j), W3(i, j) can be calculated
using (4.91)-(4.93) directly from equation (4.90):
W1(i, j) = − gyy(i − 1, j) · dt2
4µ(i − 1, j) · εy(i − 1, j) ·√g(i, j)√
g(i − 1, j)(4.91)
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 128
W2(i, j) = 1 +gyy(i, j) · dt2
4µ(i − 1, j) · εy(i, j) ·√
g(i, j)√
g(i − 1, j)
+gyy(i, j) · dt2
4µ(i, j) · εy(i, j) · g(i, j)(4.92)
W3(i, j) = − gyy(i + 1, j) · dt2
4µ(i, j) · εy(i + 1, j) · g(i, j)(4.93)
However, as far as the boundary cell (i = 1, i = xmax, j = 1 or j = ymax) need
to be calculated, equation (4.90) cannot be applied directly. So the periodic boundary
relationships (equations (4.94) and (4.95)) need to be introduced into equation (4.90).
F (1 − 1, j) = eik·(xmaxdx)x·F (xmax, j)
F (1 − 2, j) = eik·(xmaxdx)x·F (xmax − 1, j)
F (xmax + 1, j) = e−ik·(xmaxdx)x·F (1, j)
F (i, 1 − 1) = eik·(ymaxdy)y·F (i, ymax)
F (i, ymax + 1) = e−ik·(ymaxdy)y ·F (i, 1) (4.94)
G(1 − 1, j) = G(xmax, j)
G(xmax + 1, j) = G(1, j)
G(i, 1 − 1) = G(i, ymax)
G(i, ymax + 1) = G(i, 1) (4.95)
where F denotes the field value and G stands for the spatial and material parameters.
The left hand side of equation (4.94) is the field expression for the cells outside the
computational domain. So, when these values are required, their equivalent right hand
side pair which consists of value of cells inside the computational domain can be referred
to. As for the spatial and material parameters, they can be simply displaced due to the
periodicity of the EBGs. Thus the coefficient matrix W1(1, j), W2(1, j), W3(1, j) can be
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 129
calculated using equation (4.96), (4.97) and (4.98). [W1, W2, W3] (i = xmax, j = 1, j =
ymax) can be calculated in a similar way.
W1(1, j) = − gyy(xmax, j) · dt2
4µ(xmax, j) · εy(xmax, j) ·√g(1, j)√
g(xmax, j)(4.96)
W2(1, j) = 1 +gyy(1, j) · dt2
4µ(xmax, j) · εy(1, j) ·√
g(1, j)√
g(xmax, j)
+gyy(1, j) · dt2
4µ(1, j) · εy(1, j) · g(1, j)(4.97)
W3(1, j) = − gyy(1 + 1, j) · dt2
4µ(1, j) · εy(1 + 1, j) · g(1, j)(4.98)
The calculation of the the resulting matrix S can be performed in the same way. When
calculating the main area of the unit cell (2 < i < xmax and 1 < j < ymax), the entries
in the resulting matrix S(i, j) can be calculated using equation (4.99) which is readily
provided in the right hand side of equation (4.90).
S(i, j) = Dyn(i,j) − dt
2√
g(i, j)[Hn
z (i, j) − Hnz (i − 1, j)]
− dt2
4√
g(i, j)·[
Enx (i, j + 1) − En
x (i, j)µ(i, j)
√g(i, j)
− Enx (i − 1, j + 1) − En
x (i − 1, j)µ(i − 1, j)
√g(i − 1, j)
]
+dt2
16µ(i, j)g(i, j)
·⎡⎣gxy(i + 1, j)
εy(i + 1, j)
a=i+1;b=j+1∑a=i;b=j
Dxn+12 (a, b) − gxy(i, j)
εy(i, j)
a=i;b=j+1∑a=i−1;b=j
Dxn+12 (a, b)
⎤⎦
− dt2
16µ(i − 1, j)√
g(i, j)√
g(i − 1, j)
·⎡⎣gxy(i, j)
εy(i, j)
a=i;b=j+1∑a=i−1;b=j
Dxn+12 (a, b) − gxy(i − 1, j)
εy(i − 1, j)
a=i−1;b=j+1∑a=i−2;b=j
Dxn+12 (a, b)
⎤⎦ (4.99)
However, when calculating S(i, j) for (i = 1, 2, xmax and j = 1, ymax), equation (4.94)
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 130
and equation (4.95) need to be introduced into equation (4.90) when a value from outside
the computational domain is required. After all the entries are expressed using values
inside the computational domain, equation (4.90) can be presented in matrix form as:
W(j) · Dy(j)n+ 1
2 = S(j)n+ 12 (4.100)
where W is the coefficient matrix, Dy is the fields to be updated, and S is the resulting
matrix:
W(j) =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
W(i=1,j)2 W
(i=2,j)1 0 ... 0 W i=xmax,j
3
W(i=1,j)3 W
(i=2,j)2 W
(i=3,j)1 ... 0 0
0 W(i=2,j)3 W
(i=3,j)2 ... 0 0
0 0 W(i=3,j)3 ... 0 0
0 0 0 ... 0 0
... ...
0 0 0 ... W(i=xmax−1,j)1 0
0 0 0 ... W(i=xmax−1,j)2 W i=xmax,j
1
W(i=1,j)1 0 0 ... W
(i=xmax−1,j)3 W i=xmax,j
2
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
Dy(j) =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
Dy(i=1,j)
Dy(i=2,j)
Dy(i=3,j)
...
Dy(i=xmax−1,j)
Dy(i=xmax,j)
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(4.101)
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 131
S(j) =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
S(i=1,j)
S(i=2,j)
S(i=3,j)
...
S(i=xmax−1,j)
S(i=xmax,j)
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(4.102)
4.5 Validation of the ADI-NFDTD Method
In this section, the ADI-NFDTD algorithm is verified by the modelling of wave propaga-
tion in free space, in which, a much larger dt which violates the CFL condition is used. In
the second simulation, the ADI-NFDTD and the NFDTD algorithm are used in the mod-
elling of a two dimensional cylindrical perfect conductor (PEC) cavity resonator. Simula-
tion results are compared in terms of time and frequency domain, including the numer-
ical errors and the late time instability. In the third simulation, a two dimensional cylin-
drical copper cavity resonator is modelled using the ADI-NFDTD and the ADI-FDTD
methods. The numerical accuracy and the computing efficiency, including the computer
memory and computation time, are compared.
4.5.1 Removal of the CFL Stability Criteria
In this section, radio propagation in free space is modeled using both the conventional
NFDTD and the proposed ADI-NFDTD scheme. This is done as a pilot simulation to
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 132
verify that the ADI-NFDTD algorithm is free from the CFL stability criteria. The compu-
tational efficiency of the proposed ADI-NFDTD is also demonstrated.
• Parameters Of The Free Space Wave Propagation Model
The computational domain is (0.3m × 0.3m) in free space meshed by (30 × 30) cells
(figure 4.9). A point source with a continuous sine wave at 1GHz is excited at the center
of the domain (15, 15). The magnetic field (Hz) is probed at position (6, 9). Mur’s first
order absorbing boundary condition is used.
0 0.1 0.2 0.30
0.1
0.2
0.3
Figure 4.9: A mesh of the (0.3m× 0.3m) free space domain.
• Simulation Results
With a time interval dt = 1e−10s and a simulation period of 5000 time steps, the ADI-
NFDTD algorithm produces a stable output. The first 40 time steps output is plotted in
figure 4.10(a). The resolution of the Fast Fourier Transform (FFT) to the time domain
signal is 0.002GHz. The relative error of the computed operating frequency is 0.1%.
Hence the proposed ADI-NFDTD demonstrates very low numerical dispersion. How-
ever, in the conventional NFDTD algorithm, the same modelling parameters will result in
an unstable temporal output at a very early stage, shown in figure 4.10(b). This indicates
that the dt of 1e − 10s has already violated the CFL condition for the NFDTD.
By decreasing dt step by step in the simulation, dt = 5e − 12s is found to be the
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 133
maximum value which satisfies the CFL condition and gives a stable result for the same
physical period (up to 100000 time steps). The same FFT resolution with the previous
ADI-NFDTD one can be achieved (0.002GHz). The relative error rate of the frequency
spectrum is also 0.1%.
The computational efficiency of the two NFDTD schemes is also compared. With the
same FFT resolution (0.002GHz) and the numerical error rate of 0.1%, the simulation
time is compared. On a Pentium IV 2.40GHz PC with RAM of 1.5GB, under computing
environment of Matlab, the ADI-NFDTD program runs 5000 time steps at a rate of 20
time-steps/second; and the conventional NFDTD one runs 100000 time steps at a rate of
250 time-steps/second. So 250 seconds are used in the ADI-NFDTD simulation and 400
seconds are used in the conventional NFDTD one for achieving the same stability in the
same period of physical time and the same level of accuracy. The ADI-NFDTD bears a
saving factor of 1.6 in simulation time with reference of the conventional NFDTD in this
set of simulations. These results are summarized in Table 4.1.
NFDTD ADI-NFDTD
dt (second) 5e − 12 1e − 10
Number of Iteration 100000 5000
Computing time for one iteration(second)
1/250 1/20
Total Computer Run Time (second) 400 250
Saving rate of the Total ComputerRun Time by the ADI-NFDTD
- 1.6
Stability Status for the results Stable Stable
FFT resolution 0.002GHz 0.002GHz
Relative Error Rate in Frequency 0.1% 0.1%
Table 4.1: Comparison of the computational time and accuracy of the simulation results ona Pentium IV 2.40GHz PC with RAM of 1.5GB. With the same level of accuracy, the ADI-NFDTD shows a saving rate of 1.6 in total computational time compared with the conven-tional NFDTD.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 134
0 10 20 30 40-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Time steps (time interval dt = 100ps)
Ma
gn
etic F
ield
Hz (
A/m
) ADI-NFDTD
(a)
0 5 10 15 20
-2
0
2
4
6
8
10x 10
29
Time steps (time interval dt = 100ps)
Ma
gn
etic fie
ld H
z (
A/m
) conventional NFDTD
(b)
Figure 4.10: Hz field temporal results for the modelling of a single frequency wave propa-gation in free space, with time interval dt = 100ps in both the ADI-NFDTD and the NFDTDsimulations. (a) the ADI-NFDTD results; (b) the conventional NFDTD results.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 135
4.5.2 Numerical Efficiency and the Late Time Instability Improvement over
the Conventional NFDTD
In this section, the conventional NFDTD and the proposed ADI-NFDTD schemes are ap-
plied to calculate the resonant frequency of a two-dimensional cavity resonator. Numeri-
cal efficiency of the ADI-NFDTD is demonstrated and the late time instability of NFDTD
is greatly improved.
• Parameters Of The Cylindrical PEC Cavity Resonator
A cylindrical perfect electric conductor (PEC) cavity resonator is assumed to be infi-
nite long. The radius of the cavity is 0.15m. The cavity is filled with vacuum of permit-
tivity εr = 1. The outer material is PEC, which means the skin depth of the material is
small enough so that penetration of the EM field through the material can be neglected.
• The FDTD Model
Since the metallic layer enclosing the cavity is thick enough to be looked as ’infinitely’
thick from inside the cavity, the whole computational domain can be simplified into a two
layer model, with an outside metallic layer terminated at an adequate thickness (much
greater than the skin depth). The computational region is set to be (0.52m × 0.5333m)
meshed by (25 × 27) cells (shown in figure 4.11).
A sine modulated cosine pulse (equation( 4.103)) is excited at selected source posi-
tions inside the cavity (see figure 4.12) to provide a wide band excitation to excite all the
possible TE modes:
Source(n) = Am · sin(2πf · n · dt) · (1 − cos(2πf · n · dt)) (4.103)
where Am relates to the amplitude of the signal, f is the frequency parameter, dt is the
time increment and 1 ≤ n ≤ 1/(dt · 2πf) is the iteration index. Figure 4.13 shows an
example of the pulse when Am=1, f = 7GHz and dt = 1ps.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 136
After a period of time, the modes that are not supported by the cavity will vanish
and only those supported by the cavity will resonate. Then temporal signatures at sam-
pled probing points will go through a Fast Fourier transformation (FFT) and the resonant
modes will be found as peaks in frequency spectra. The position of the probing points is
also illustrated in figure 4.12.
0.1 0.2 0.3 0.4 0.5
0.1
0.2
0.3
0.4
0.5
Figure 4.11: The mesh of the cut plane of the cylindrical cavity with radius r = 0.15m, withthe air-metal boundary indicated by the red line.
Figure 4.12: The NFDTD modelling of the cavity.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 137
0 1 2 3 4 5
x 10−9
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Am
plitu
de
(a)
0 2 4 6 8 10
x 109
0
20
40
60
80
100
Frequency (Hz)
Mag
nitu
de
(b)
Figure 4.13: An example of the excitation signal in time and frequency domain. Am=1, f =7GHz and dt = 1ps.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 138
• Simulation Results
The magnetic fields inside the resonator at the same probe location modelled by the
ADI-NFDTD and the NFDTD schemes are compared. Both simulations use the same
mesh profile (figure 4.11) and timestep dt = 1ps. It is expected that the ADI-NFDTD
results are less accurate than the conventional NFDTD ones. The first 8000 time steps
results calculated from both algorithms are normalized by their own peaks and are com-
pared in figure 4.14. The resemblance of the two curves can be taken as one of the verifi-
cations of the proposed ADI-NFDTD algorithm.
Figure 4.14: The comparison of the normalized H field time domain results of the first8000 time steps (dt = 1ps) from the conventional NFDTD (solid line) and the ADI-NFDTDschemes (dotted line).
The comparison of the ADI-NFDTD and the NFDTD schemes shows that the late
time instability occurs much later in the ADI-NFDTD scheme than in the conventional
NFDTD one. For instance, in the conventional NFDTD simulation using the dt of 1ps,
the result began to become unstable after 12000 time steps, which corresponds to 12ns
in physical time (details can be found in figure 4.15(a)). After that, the energy in the
simulation domain will increase exponentially due to the late time instability ( figure
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 139
4.15(b)). The ADI-NFDTD result is stable until 100000 time steps, which corresponds to
100ns in physical time ( figure 4.15(c)).
More simulations were carried out using different dt in order to investigate the sta-
bility of the proposed ADI-NFDTD scheme. It is shown that unlike the conventional
ADI-FDTD, the proposed ADI-NFDTD scheme is not an unconditionally stable scheme.
Instability in the ADI-NFDTD temporal results is observed. The CFL stability condition
on dt in the conventional NFDTD is also observed in this sets of simulations. When
dt > 16ps, the result becomes unstable at a very early stage. In other words, as soon
as the energy propagates to the cell which requires the smallest ∆t ij , dt does not satisfy
the local CFL condition anymore and causes instability immediately. However, with the
ADI-NFDTD, this instability does not occur with the same or greater dt value.
In order to study the relationship of the accuracy of the simulated results with the
time interval dt used in the simulation, the resonant frequencies calculated from FFT are
compared with the theoretical resonant frequencies. For those ADI-NFDTD results which
become unstable in the later time steps, the data in their stable period are used. There is
a tradeoff when considering the amount of data to be used. If only the data in the stable
period are used, the frequency resolution in FFT will be low limiting the accuracy of the
frequency result. In this case, some data at the beginning of the unstable period will have
to be involved. The more unstable data are used, the higher frequency resolution can be
achieved after FFT. However, too many data in the unstable period will result in a high
noise level in the frequency spectra. As is mentioned previously, for the dt value of 1ps,
the result of the conventional NFDTD becomes unstable from around 12000 time step.
After that, the energy of the field increases gradually and the number of the NFDTD
temporal samples used for FFT are chosen to be 40000 to balance the aforementioned
tradeoff. The number of the ADI-NFDTD temporal samples is chosen to be 100000. So
the frequency resolution of the spectra for the conventional NFDTD and the ADI-NFDTD
are 0.025GHz and 0.01GHz respectively.
The calculated resonant frequency spectra with dt = 1ps is plotted in figure 4.16. The
resonant frequencies are compared with the theoretical results in Table 4.2. The absolute
value of the difference of the calculated results with the theoretical results[151] can be
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 140
0 0.5 1 1.5 2 2.5 3
x 104
-15
-10
-5
0
5
10
15
Magnetic f
ield
Hz(A
/m)
Time step (dt = 1ps)
(a)
0 1 2 3 4 5
x 104
-5000
0
5000
Time steps ( time interval: 1e-12s)
Magnetic fie
ld H
z
conventional NFDTD
(b)
0 2 4 6 8 10
x 104
-15
-10
-5
0
5
10
15
Time steps ( dt = 1ps)
Magnetic f
ield
Hz (
A/m
)
(c)
Figure 4.15: Hz field temporal results with dt = 1ps. (a) the first 30000 time steps by theNFDTD; (b) the first 50000 time steps by the NFDTD. (c) the first 100000 time steps by theADI-NFDTD.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 141
firstly obtained. Then it is divided by the theoretical value and expressed in percentage
form giving the relative error rate (RER) of each resonant mode. Finally, the standard
deviation of the calculated resonant frequencies for all the resonant modes (from 0GHz
to 2.732GHz) are taken as a parameter which will be referred in the following as the
’averaged relative error rate (ARER)’. (Note that the theoretical frequency is used as the
expected value in the standard deviation calculation, and this applies to all the error rate
calculations in this chapter.)
0 0.5 1 1.5 2 2.5 3
x 109
0
1000
2000
3000
4000
5000
Frequency (Hz)
Mag
netic
fiel
d H
z(A
mpe
r/m
eter
)
NFDTDADI−NFDTD
Figure 4.16: Resonant frequency spectra of the cavity resonator calculated from the twoschemes, with dt = 1ps. Solid line: the conventional NFDTD scheme; dotted line: the ADI-NFDTD scheme.
As can be seen in figure 4.16, the accuracy of the conventional NFDTD is limited by
this frequency resolution while by using the ADI-NFDTD scheme, the resonant frequency
can be located in the frequency spectra with a higher resolution. Some resonant modes
have very close frequencies. For example, it is hard to distinct different resonant modes
at around 2.5GHz by the conventional NFDTD simulation while it is easy to do so in
the ADI-NFDTD one with a higher frequency resolution. It can also be seen that the
noise level of the ADI-NFDTD result is lower than the conventional NFDTD one. That
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 142
is because the time domain data from the former include less unstable numerical errors
Averaged Relative Error Rate (ARER)from all the Genuine Modes 2.85 - 2.85ARER from all the Modes 4.70 - 4.75
Table 4.2: Comparison of the accuracy of the resonant modes of the aforementioned PECresonator calculated by the conventional NFDTD and the ADI-NFDTD with the theoreticalresults. In both simulations, dt = 1ps. The modes with ∗ are spurious modes introducedin the meshing of the cylindrical cavity. They are compared with the nearest genuine modewhen RER is evaluated and hence they cause big numerical errors. In the first deviation ofRER calculation, these errors are not included. In the second deviation of RER calculation(with ∗), these errors are taken into consideration.
In ADI-NFDTD, it is observed that the high frequency components will attenuate with
the increment of the time step dt used. So with big dt, the high frequency components be-
come hard to distinguish among the noise. This is illustrated in figure 4.17, in which the
frequency spectra calculated from the ADI-NFDTD result with dt = 17ps are compared
with the spectra from the conventional NFDTD with dt = 1ps. Because of the attenuation
in high frequency in ADI-NFDTD, the amplitude of the spectra of ADI-NFDTD result is
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 143
multiplied uniformly by a factor of 3.5 to give a better view of comparison with the con-
ventional NFDTD spectrum result. It can be seen that the lower frequency components
can still be precisely evaluated in ADI-NFDTD results; and the higher frequency compo-
nents also can be identified by the trends but not as precisely as in a smaller dt case (e.g.
figure 4.16 in which dt = 1ps).
So in the ADI-NFDTD scheme, the choice of the dt value depends largely on the re-
quirement on accuracy. It is worth noticing that this dt value of 17ps used in this ADI-
NFDTD simulation has violated the CFL condition of the conventional NFDTD. That
means the same dt, in conventional NFDTD scheme, will result in a temporal result un-
stable at a very early stage and no reasonable frequency spectrum is possible to obtain.
The averaged relative error rate of the resonant frequency by the proposed ADI-
NFDTD scheme and the conventional NFDTD scheme are plotted against the relative
time interval in figure 4.18. Theoretically, the ADI-NFDTD may not necessarily provide
better accuracy than the conventional NFDTD.
However, as far as this set of simulations are concerned, the ADI-NFDTD scheme
can always provide longer stable temporal results to perform the FFT with a frequency
resolution which is small enough. So with some dt, the ADI-NFDTD result shows smaller
relative error rate. It can be seen that from the 16th unit, the relative error rate of the
conventional NFDTD results increases dramatically. That is the dt value required by the
CFL stability condition. From that value, the resonant frequencies become undetectable
with the conventional NFDTD, while in the ADI-NFDTD, the error rate only increases
slowly with the increase of the dt.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 144
0 0.5 1 1.5 2 2.5 3
x 109
0
1000
2000
3000
4000
5000
Frequency (Hz)
Mag
netic
fiel
d H
z(A
mpe
r/m
eter
)
NFDTDADI−NFDTD
Figure 4.17: Resonant frequency spectrum of the cavity resonator calculated from temporalresults by the ADI-NFDTD with dt = 17ps (dotted line), compared with the spectrum cal-culated from temporal results by the conventional NFDTD with dt = 1ps (solid line). Theamplitude of the ADI-NFDTD result is uniformly amplified by 3.5 times.
0 5 10 15 16 202
3
4
5
6
7
8
Relative Time Interval dt ( in the unit of 1ps)
Rel
ativ
e E
rror
Rat
e (%
)
NFDTDADI−NFDTD
Figure 4.18: The averaged relative error rate of the NFDTD and the ADI-NFDTD simulatedresonant frequency spectra as the function of relative time interval dt/∆t(∆t = 1ps). Whendt > 16∆t, the CFL condition is violated. The direct consequence is that the relative errorrate of the conventional NFDTD increases to 100%, while ADI-NFDTD still provides resultswith a relative error rate of less than 5%.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 145
4.5.3 Numerical Efficiency Improvement over the Orthogonal ADI-FDTD
In this section, a two-dimensional cavity resonator is modelled using both the conven-
tional Yee’s FDTD and the proposed ADI-NFDTD scheme to demonstrate the numerical
efficiency improvement of the novel ADI-NFDTD over the conventional orthogonal ADI-
FDTD.
• Parameters Of The Cylindrical Copper Cavity
A cylindrical copper cavity resonator is assumed to be infinitely long. The radius
of the cavity is 0.15m. The cavity is filled with vacuum of permittivity εr = 1. The
outer material is chosen to be copper, with relative permittivity εr = 1 and conductivity
σ = 5.8 × 107 S/m. The computational domain to be modelled is (0.6m × 0.6m).
• The FDTD Model
The modelling procedure is quite similar to that in 4.5.2 where the ADI-NFDTD and
the NFDTD are compared by the modelling of a PEC resonator. Only in this modelling,
the mesh files (figures 4.20 - 4.23) are different and the cavity is excited by a wide band
modulated Gaussian pulse (equation (4.104)). Once the temporal responses of the probe
points go through the FFT, the resonant frequencies are identified as peaks in the spectra
and they are compared with the theoretical results. Figures 4.20, 4.21, 4.22 and 4.23 show
the meshes of the ADI-NFDTD and the ADI-FDTD model under different spatial resolu-
tions. The mesh size are (46×46) for the ADI-NFDTD modelling, (60×60), (70×70), and
(80×80) for the ADI-FDTD ones respectively. The time step is chosen to be 2ps and 25000
iterations are run in each simulations. This ensures the same FFT resolution (0.02GHz) in
every modelling.
Source(n) = Amej2πf ·n·dt · e−((n−delay)·dt
τ2)2 (4.104)
where Am is the maximum amplitude, f is the operating frequency, delay = 1dt·f is the
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 146
delay in time of the pulse, τ2 = 18f is the pulse half-duration at the 1/e points, dt is the
time increment and n is the iteration index. In this simulation, Am = 1, f = 0.5GHz,
dt = 2ps and the modulated Gaussian pulse is plotted in time domain and frequency
domain in figure 4.19.
0 1 2 3 4 5 6 7 8 9 10
x 10−9
0
0.05
0.1
time (s)
Am
plitu
de
real part of the excitation functionimaginary part of excitation function
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 109
0
5
10
15
20
frequency (Hz)
Mag
nitu
de
(b)
Figure 4.19: The modulated Gaussian pulse as excitation in (a) time domain and (b) fre-quency domain.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 147
0 0.1 0.2 0.3 0.4 0.5 0.60
0.1
0.2
0.3
0.4
0.5
Figure 4.20: In ADI-NFDTD, the computation domain is meshed by a (46 × 46) grid.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 148
10 20 30 40 50 60
10
20
30
40
50
60
Figure 4.21: In ADI-FDTD, the computation domain is meshed by a (60 × 60) grid.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 149
10 20 30 40 50 60 70
10
20
30
40
50
60
70
Figure 4.22: In ADI-FDTD, the computation domain is meshed by a (70 × 70) grid.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 150
10 20 30 40 50 60 70 80
10
20
30
40
50
60
70
80
Figure 4.23: In ADI-FDTD, the computation domain is meshed by a (80 × 80) grid.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 151
• Simulation Results
Table 4.3 tabulates the resonant frequencies obtained from the FFT of the temporal
results calculated by the ADI-NFDTD with mesh (46 × 46). Table 4.4, 4.5, and 4.6 show
the resonant frequencies calculated from the ADI-FDTD with mesh (60 × 60), (70 × 70),
and (80 × 80) respectively.
The relative error rate of each mode is calculated by the absolute value of the dif-
ference of the calculated frequency with the theoretical one[151] divided by the latter in
expression of percentage. Then the standard deviation of the relative error rates from all
the frequencies (from 0GHz to 3GHz ) are taken as a measure of an overall relative error
rate.
The frequency spectra calculated from the ADI-FDTD with different spatial resolu-
tions and the ADI-NFDTD are compared in figures 4.24, 4.25 and 4.26. The theoretical
results are denoted by dashed lines vertical to x-axis at their designated frequencies.
Then the numerical error including the overall relative error rates and the number of
spurious modes from the ADI-NFDTD and the ADI-FDTD modellings are summarized
in table 4.7. The computational effort including the computer memory employed in the
simulation and the computation time for each modelling are also compared in this table.
It can be clearly seen from this table that to model the same structure with the same dt and
iteration number (and the same FFT resolution) the ADI-NFDTD method can provide
results with less numerical errors using less computer memory and computation time
compared with the ADI-FDTD method. This comparison demonstrates the accuracy and
computing efficiency of the ADI-NFDTD over the ADI-FDTD method.
However, compared to the ADI-FDTD method, the ADI-NFDTD method is not an
unconditionally stable method. When the time step is small, the temporal result still
suffers the late time instability which is inherited from the NFDTD scheme. This is the
drawback of the ADI-NFDTD method.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 152
Theoretical The Genuine Relative Error The Spurious Relative ErrorResults [151] Modes Rate (RER) Modes Rate (RER)
Averaged Relative Error RateIncluding Only The Genuine 3.04 - -Modes (ARER-G)Averaged Relative Error RateIncluding All the Modes - - 4.25(ARER)
Table 4.3: Comparison of the accuracy of the resonant modes of the aforementioned copperresonator calculated by the ADI-NFDTD (Mesh 46 × 46) with the theoretical results.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 153
Theoretical The Genuine Relative Error The Spurious Relative ErrorResults [151] Modes Rate (RER) Modes Rate (RER)
Averaged Relative Error RateIncluding Only The Genuine 3.89 - -Modes (ARER-G)Averaged Relative Error RateIncluding All the Modes - - 6.22(ARER)
Table 4.4: Comparison of the accuracy of the resonant modes of the aforementioned cop-per resonator calculated by the conventional ADI-FDTD (Mesh 60 × 60) with the theoreticalresults.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 154
Theoretical The Genuine Relative Error The Spurious Relative ErrorResults [151] Modes Rate (RER) Modes Rate (RER)
Averaged Relative Error RateIncluding Only The Genuine 3.87 - -Modes (ARER-G)Averaged Relative Error RateIncluding All the Modes - - 4.52(ARER)
Table 4.5: Comparison of the accuracy of the resonant modes of the aforementioned cop-per resonator calculated by the conventional ADI-FDTD (Mesh 70 × 70) with the theoreticalresults.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 155
Theoretical The Genuine Relative Error The Spurious Relative ErrorResults [151] Modes Rate (RER) Modes Rate (RER)
Averaged Relative Error RateIncluding Only The Genuine 3.46 - -Modes (ARER-G)Averaged Relative Error RateIncluding All the Modes - - 5.92(ARER)
Table 4.6: Comparison of the accuracy of the resonant modes of the aforementioned cop-per resonator calculated by the conventional ADI-FDTD (Mesh 80 × 80) with the theoreticalresults.
Computer Memory (MB) 84.7 105.5 132.1 179.2Computer Run Time (minutes) 50 123 204 294
Table 4.7: Comparison of the accuracy and computer resources ( computer memory andcomputing time) of the ADI-NFDTD and ADI-FDTD in calculating the resonant modes of thecopper cavity. The simulations are run under programming environment of Matlab 2007b.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 156
Figure 4.26: The resonant frequency spectra calculated by the ADI-NFDTD (with mesh by 46by 46 grids) and the ADI-FDTD (with mesh by 80 by 80 grids).
4.5.4 Discussions
In this section, the proposed ADI-NFDTD is compared with the conventional NFDTD
scheme by modelling the free space wave propagation and a resonanting PEC cylindri-
cal cavity. The ADI-NFDTD scheme is also compared with the ADI-FDTD algorithm
by modelling a copper cylindrical cavity. The observations from these simulations are
discussed below.
• Stability Characteristics
The simulation results firstly show that the CFL condition of the conventional NFDTD
algorithm is removed by using the ADI-NFDTD scheme. The chosen value of the time
step dt is only limited by the accuracy requirement.
Unlike the conventional ADI-FDTD algorithm, the ADI-NFDTD scheme is not an un-
conditionally stable algorithm. This algorithm suffers the late time instability. However,
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 158
compared with the conventional NFDTD simulation with the same dt, the unstable re-
sult occurs at a later stage. That means the late time instability of the NFDTD is largely
improved by the use of the ADI technique.
• Accuracy
Compared with the NFDTD results, the ADI-NFDTD time domain results suffer at-
tenuation in amplitude with time going and a delay in phase. Increasing the time interval
dt results in a higher attenuation speed and a larger phase delay. The choice of dt will af-
fect the accuracy of the frequency spectrum. When a smaller dt is used, the ADI-NFDTD
is able to provide a more precise frequency spectrum with a high FFT resolution and a
low noise level. However, with an increase of dt, the amplitude attenuation in high fre-
quency components will increase. Consequently, higher frequency components tend to
be buried by the noise in the frequency spectra. So the accuracy in the higher frequency
band will decrease with the increase of dt, although the frequency resolution remains
high.
Compared with the ADI-FDTD method, the ADI-NFDTD method does not employ
the staircase approximation. The curved or oblique structures can be modelled confor-
mally, which means less grids can be applied and higher accuracy can be obtained in the
ADI-NFDTD modelling than in the ADI-FDTD one.
• Computational Efficiency
As a larger dt can be used in the ADI-NFDTD simulation than in a conventional
NFDTD one, the former can be more computationally efficient than the latter. However,
this increase in efficiency may be limited with the consideration of the accuracy issue
with a relatively large dt. The mesh file is another factor that affects the improvement on
efficiency in the ADI-NFDTD scheme. The improvement may be more significant with
those kinds of meshes that have locally very small and/or distorted cells.
Compared with the ADI-FDTD method, the ADI-NFDTD also demonstrated an im-
proved computational efficiency. As a direct result of coarser grids, computational effort
in terms of computer memory and computation time can be saved.
Chapter 4 The Development of the Alternating-Direction Implicit Nonorthogonal FDTD Method 159
4.6 Summary
In this chapter, two extensions of FDTD, namely the nonorthogonal FDTD (NFDTD) and
the ADI-FDTD are briefly reviewed. A novel two-dimensional ADI-NFDTD in curvilin-
ear coordinate system that is free of the CFL stability condition is then presented. This is
followed by presenting three sets of numerical simulations that served as validations of
the ADI-NFDTD method.
1. The CFL stability condition is proven to be removed in the ADI-NFDTD method.
This is done by the free space wave propagation modelling using both the ADI-NFDTD
and the NFDTD. Therefore the time step dt in the ADI-NFDTD simulation is no longer
restricted by the numerical stability.
2. Comparison of ADI-NFDTD with the NFDTD: These two methods are compared
when modelling a PEC cavity resonator. It is demonstrated that the ADI-NFDTD tem-
poral result resembles the NFDTD result when dt is small. When a larger dt is used, the
error on the amplitude and phase of the temporal results will increase but the late time
stability of the NFDTD scheme is significantly improved. Since the dt is no longer re-
stricted by the CFL condition, the computational efficiency can be improved by the use
of a larger dt. This improvement is more significant if the mesh contains locally very
small and/or distorted cells.
3. Comparison of ADI-NFDTD with the ADI-FDTD: Since an oblique or curved sur-
face is modelled conformally instead of employing the staircase approximation, the re-
quirement of the spatial resolution in the ADI-NFDTD modelling is considered to be
lower than that in the ADI-FDTD modelling. This is verified by the modelling of a cop-
per cavity resonator using these two methods. The numerical error ( including relative
error rate and the number of spurious modes) and the computational resource (includ-
ing the computer memory and the CPU time) are compared. The ADI-NFDTD scheme
demonstrated a better accuracy and a saving on the computer resource. However, the
ADI-NFDTD scheme is not unconditionally stable like the ADI-FDTD method. It does
suffer the late time instability.
Chapter 5
NFDTD and ADI-NFDTD modelling
of EBG Structures
In this chapter, numerical experiments for modelling two-dimensional structures with
curved surfaces using NFDTD method will be reported. The comparison of the NFDTD
and Yee’s FDTD algorithm in modelling EBG structures is done in the first two simu-
lations: the calculation of the dispersion diagram of an infinite EBG structure and the
transmission coefficient calculation of a (semi-)finite bandgap structure. From these com-
parisons, the efficiency and accuracy of the NFDTD method is demonstrated. NFDTD
modelling of waveguide mode observed from a defect EBG structure and a prism shaped
EBG-like refractor are also presented.
In the last section of this chapter, numerical performance of the proposed ADI-NFDTD
method is demonstrated by modelling an EBG unit cell with curved inclusions. With a
reduction in the late time instability demonstrated by the ADI-NFDTD method, the ADI-
NFDTD results are shown with a high frequency resolution and a low noise level.
160
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 161
5.1 NFDTD Modelling of the Infinite Electromagnetic Bandgap
Structures
In this section, two-dimensional (2-D) infinite EBG structures with curved inclusions are
studied numerically by both the NFDTD and the Yee’s FDTD schemes. By comparing
the numerical accuracy of the two schemes, the requirement on spatial resolutions, com-
puter memory, processing time, the efficiency and accuracy of the NFDTD method is
demonstrated. In this study, the unit cell approach that was previously discussed was
employed.
5.1.1 The Model of the EBG Structures
Metallic rods periodically loaded in square and triangular/rhombic lattices in free space
are modelled using both the NFDTD and the Yee’s schemes. These EBG structures are
infinite in x and y direction with a lattice constant (period) a. In the z direction, the rods
are infinitely long. Each rod is made from copper with relative permittivity εr = 1 and
conductivity σ = 5.8 × 107 S/m. The ratio of the radius r to the lattice constant a is cho-
sen to be r/a = 0.2. Numerical simulations are performed to determine the dispersion
diagrams for both the transverse electric (TE) and transverse magnetic (TM) polariza-
tion in square (figure 5.1) and triangular/rhombic lattices (figure 5.2). Results from both
the NFDTD and Yee’s FDTD methods are compared in terms of accuracy, efficiency and
robustness.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 162
(a)
Γ X
M
a
r
unit cell
(b)
Figure 5.1: (a) The cylindrical metallic rods in free space in square lattice with radius r,spacing a and r/a = 0.2. (b) The x-y cut plane of the EBGs. The unit cell is the area within thedashed red line. The Brillouin Zone in the reciprocal lattice is shown in the light blue area.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 163
(a)
Γ
X J
a r
unit cell
unit cell
unit cell
(b)
Figure 5.2: (a) The cylindrical metallic rods in free space in triangular/rhombic lattice withradius r, spacing a and r/a = 0.2. (b)The x-y cut plane of the EBGs. The unit cell is the areawithin the dashed red line. The Brillouin Zone in the reciprocal lattice is shown in the lightblue area.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 164
5.1.2 Mesh of the Unit cell
The unit cells in square and triangular/rhombic lattice (see figure 5.1(b), figure 5.2(b)) are
meshed with different spatial resolutions using Yee’s scheme and the NFDTD scheme,
with examples shown in figure 5.3 and figure 5.4.
5 10 15 20 25 30
5
10
15
20
25
30
(a) (b)
10 20 30 40 50
10
20
30
40
50
20 40 60 80
10
20
30
40
50
60
70
80
(c) (d)
Figure 5.3: Examples of the mesh schemes for an unit cell from the square lattice in theNFDTD and the Yee’s FDTD modelling. (a) (18 × 18) NFDTD cells. (b) (30 × 30) FDTD cells.(c) (50 × 50) FDTD cells. (d) (80 × 80) FDTD cells.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 165
(a) (b)
(c) (d)
Figure 5.4: Examples of the mesh schemes for an unit cell from the triangular lattice in theNFDTD and the Yee’s FDTD modelling. (a) (18 × 15) NFDTD cells. (b) (30 × 26) FDTD cells.(c) (50 × 42) FDTD cells. (d) (80 × 69) FDTD cells.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 166
5.1.3 Settings in the FDTD Simulations
Because the rods in the EBGs are periodically located and infinite in the x and y directions,
the whole structure can be studied by investigating only one unit cell and by applying pe-
riodic boundary conditions (PBC). The excitation, the probe settings, the computational
domain and the boundary conditions are illustrated in figure 5.5.
(a)
(b)
Figure 5.5: The modelling schemes of the unit cell approach for EBG structures. (a) for squarelattice of figure 5.1; (b) for triangular/rhombic lattice of figure 5.2.
A modulated Gaussian pulse is used to provide a wide band excitation at different
positions inside the unit cell domain. These source points, as well as the probe points
introduced in the following, are randomly chosen but need to excite and cover all the
possible modes inside the structure. After all the possible modes are excited by the wide
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 167
band excitation, most modes will die out soon as evanescent waves. Only the modes
that are supported by the EBGs as propagating modes will propagate and hit the unit
cell again and again, resulting in accumulated energy, which can be detected as peaks
in the frequency spectra in the post processing. Temporal signals at the aforementioned
selected probe positions are used to detect all the possible transmission modes. With
the Fast Fourier Transformation (FFT) applied to these temporal signature, the frequency
spectra can be obtained and dispersion diagrams can be plotted.
As a direct solution of Maxwell’s equations, the FDTD method is an accurate time do-
main method in EM modelling. However, this accuracy depends on the spatial resolution
used in the simulation. To achieve an accurate, convergent result, the spatial resolution
should has an appropriate value. This means that the mesh should be fine enough and
a small spatial increment (∆x,∆y) will consequently require a small time increment (∆t)
by the CFL stability condition.
On the other hand, the spatial resolution and the corresponding time increment (∆t)
directly related to the computer memory and the computational time required in the sim-
ulation. These two factors are the main concerns and restrictions of the FDTD method. In
this way, a low spatial resolution is preferred in order to save computer resources (mem-
ory and time) and to enable electrically large models to be simulated. As a consequence,
it is interesting to investigate the relationship of the accuracy/convergency of the results
with the spatial resolution used in the simulation.
5.1.4 Simulation Results
The simulation is performed with various mesh densities and operating frequencies. The
dispersion diagrams of the aforementioned EBGs calculated by the NFDTD and Yee’s
FDTD method are compared in figures 5.6-5.9.
The simulation results demonstrate that the dispersion diagrams are converged when
the spatial resolution is adequate. The results obtained with high spatial resolutions (e.g.
a FDTD grid with spatial increment equals to one-eightieth of the operating wavelength
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 168
in free space dxY EE = λcfs
80 ) can be considered to be accurate and are taken as a reference
to validate the other FDTD schemes.
It is further noticed that with adequate spatial resolutions, results from the NFDTD
and Yee’s schemes exhibit similar error rates. Take the simulation result of EBGs with a
rectangular lattice as an example. It can be seen that the maximum error of the results
obtained from Yee’s FDTD with low spatial resolution occur near M and Γ point in the
dispersion diagram. In Table 5.1, a comparison of numerical results from various FDTD
schemes is demonstrated for the first few modes in figure 5.6 and figure 5.7.
0
0.5
1
1.5
2
Fre
quen
cy (ω
a/(2
πc))
Copper rods (r/a=0.2) in Square Lattice
Γ X M Γ
TE Mode Yee
80 80X80 NFF=1 (λ
cfs/80)
TE Mode Yee30
30X30 NFF=1 (λcfs
/30)
TE Mode NFDTD18
18X18 NFF=1(λcfs
/18)
Figure 5.6: The dispersion diagram of the first few TE modes, for the copper rods in freespace in square lattice. The Yee’s FDTD simulation result with high spatial resolution isplotted in purple line as a reference result. The Yee’s FDTD and the NFDTD result withthe low spatial resolution is plotted to compare with the high spatial resolution result. Itcan be seen that the maximum disagreement appears at M point. So the error rate at Mpoint is listed in table5.1(a). (NFF: Normalized central Feeding Frequency parameter of themodulated Gaussian pulse. It is a frequency parameter in the modulated Gaussian pulse.)
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 169
0
0.5
1
1.5
2
Fre
quen
cy (ω
a/(2
πc))
Copper rods (r/a=0.2) in Square Lattice
Γ X M Γ
TM Mode YEE
80 NFF=1 (λ
cfs/80)
TM Mode YEE30
NFF=1 (λcfs
/30)
TM Mode NFDTD18
NFF=1 (λcfs
/18)
Figure 5.7: The dispersion diagram of the first few TM modes, from the copper rods in freespace in square lattice. The Yee’s FDTD simulation result with high spatial resolution isplotted as a reference result. The Yee’s FDTD and the NFDTD result with the low spatialresolution is plotted to compare with the high spatial resolution result. It can be seen thatthe maximum disagreement appears at M point. So the error rate at M point is listed intable5.1(b). NFF: Normalized central Feeding Frequency parameter of the modulated Gaus-sian pulse. It is a frequency parameter in the modulated Gaussian pulse.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 170
0
0.5
1
1.5
2
Dispersion Diagram of the Copper Rods in Triangular Lattice
Γ X J Γ
Fre
quen
cy (ω
a/(2
πc))
TE mode Yee80
NFF=1 (λcfs
/80)
TE mode Yee28
NFF=1 (λcfs
/28)
TE mode NFDTD18
NFF=3 (λcfs
/6)
Figure 5.8: The dispersion diagram of the first few TE modes, from the copper rods in freespace in triangular/rhombic lattice. The Yee’s FDTD simulation result with high spatial res-olution is plotted in blue line as a reference result. The Yee’s FDTD result with the minimumspatial resolution is plotted in red. The NFDTD result with the minimum spatial resolutionis plotted in green.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 171
0
0.5
1
1.5
2
Γ X J Γ
Copper rods (r/a=0.2) in Triangular Lattice
Fre
quen
cy (ω
a/(2
πc))
TM Mode YEE80
80X69 NFF=1 (λcfs
/80)
TM Mode YEE30
30X26 NFF=1 (λcfs
/30)
TM Mode NFDTD18
18X15 NFF=1 (λcfs
/18)
Figure 5.9: The dispersion diagram of the first few TM modes, from the copper rods in freespace in triangular/rhombic lattice. The Yee’s FDTD simulation result with high spatial res-olution is plotted in blue line as a reference result. The Yee’s FDTD and the NFDTD resultwith the low spatial resolution is plotted to compare with the high spatial resolution result.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 172
Results by the Results by the Relative Error Rate (RER)TE high spatial low spatial approximated by the
resolution (HSR) resolution (LSR) difference of HSR and LSR.Mode
Table 5.1: Comparison of the relative error rate on the dispersion diagrams at M point for(a) TE mode, (b) TM mode, calculated by the Yee’s FDTD and the NFDTD with low spatialresolutions. This error rate is approximated by the difference of the low-spatial-resolutionresult compared with the high-spatial-resolution result.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 173
However, the NFDTD scheme exhibits an overwhelming advantage when the mini-
mum spatial resolutions required by these two algorithms are compared. Take the mod-
elling of the copper rods in a triangular lattice (TE mode calculation) as an example. Fre-
quency spectra at k = OΓ (Γ point on the dispersion diagram) is calculated by NFDTD
method using different density of meshes (size: 18× 15, 18× 15 and 10× 8) and different
normalized central feeding frequencies (NFF = 1, 3, 2). These parameters corresponds
to spatial resolutions of ( λcfs
18 , λcfs
6 and λcfs
5 ) respectively (see figure 5.10).
Results with dxNFDTD = λcfs
18 are used as a reference as it has been proved to be
adequate to provide converged and accurate results. It can be seen that dxNFDTD = λcfs
6
is the minimum spatial resolution required by NFDTD for this model to provide a correct
dispersion relation with all the genuine modes identified. Any reduction on the spatial
resolution upon this value will result in wrong dispersion relationship due to the fact that
strong spurious modes are mixed with the genuine modes in the calculated frequency
spectra.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 174
0 0.5 1 1.5 20
10
20
30
40
50
60
70
80
Normalized Frequency (ωa/2πc)
Mag
netic
Fie
ld H
z (A
/m)
Frequency Spectra of wave vector k at Γ pointCopper rods (r/a=0.2) in Triangular Lattice
Figure 5.10: The TE mode frequency spectra of the wave vector k at Γ point, for the modellingof the cylindrical copper rods in triangular lattice. The green line and the pink line showthe frequency spectra calculated by NFDTD using a high spatial resolution (dxNFDTD =λcfs
18 ) and a low spatial resolution (dxNFDTD = λcfs
6 ) respectively. The positions of peaks inthese two lines agree well, indicating all the modes founded by the low spatial resolution(dxNFDTD = λcfs
6 ) are all genuine. The blue line shows the frequency spectra using spatialresolution of dxNFDTD = λcfs
5 . On the blue line, spurious modes (marked by the dotted redcircles) are strong enough to compete with a genuine mode (marked by the dashed orangecircle) , which will cause a wrong dispersion diagram. The comparison of these three linesindicates that dxNFDTD = λcfs
6 is the minimum spatial resolution required by NFDTD forthis modelling.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 175
Figure 5.11 shows that Yee’s FDTD simulation with dx = λcfs
26 provides a result with
spurious spectrum in low frequency band. It is found from more simulations that ( λcfs
28 )
is the mininum spatial resolution required by the Yee’s scheme for this model.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
10
20
30
40
50
60
70
80
90
100
Normalized Frequency (ωa/2πc)
Mag
netic
Fie
ld H
z (A
/m)
Frequency Spectra of wave vector k at Γ point Copper rods (r/a=0.2) in Triangular Lattice TE mode
NFDTD 18X15 NFF=1 (λ/18)
Yee’s FDTD 26X23 NFF=1 (λ/26)
Figure 5.11: The TE mode frequency spectra of the wave vector k at Γ point of the cylindricalcopper rods in triangular lattice. Yee’s FDTD results with a spatial resolution of dx = λcfs
26provides a result with spurious energy in low frequency band (marked by the dotted redcircles) which are strong enough to compete with a genuine mode (marked by the dashedorange circle). The NFDTD results with a spatial resolution of dx = λcfs
18 which provide allthe genuine modes is used as a reference.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 176
More examples are shown in order to compare the minimum spatial resolutions re-
quired by the NFDTD and the Yee’s algorithms. Figure 5.12 illustrates the frequency
spectra at Γ for TE mode modelling of the copper rods in a square lattice, calculated by
using the NFDTD method with different spatial resolutions. Figures 5.13 - 5.15 show the
spectra calculated using Yee’s FDTD scheme with different spatial resolutions.
It can be seen from figures 5.12, 5.13 and 5.14, although some spurious resonances
occur at some probe positions (marked by the dotted red circles), the amplitude of them
averaged by all the probes are much smaller than that of a genuine mode. Since the
dispersion diagram is calculated by an average of frequency spectra of all the probe po-
sitions, these spurious modes can be numerically filtered out as noise and will not affect
the dispersion diagram calculation. This is referred to in this thesis as the ”system tol-
erance”. However, as shown in figure 5.15, if the spatial resolutions are reduced further,
the spurious modes become as strong as a genuine one. This means the spatial reso-
lution is beyond the system tolerance and consequently results in inaccurate dispersion
calculations for EBGs.
Figure 5.16 compares the TM mode frequency spectra for Γ point in a square lattice of
EBGs, calculated by the NFDTD and the Yee’s FDTD algorithms.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 177
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
1
2
3
4
5
6
7
8
Normalized Frequency (ωa/(2πc))
Mag
nitu
de o
f Mag
netic
Fie
ld H
z (A
/m)
Frequency Spectra for wave vector k from the center of the Brillioun Zone to Γ point (TE) (NFF: Normalized Central feeding frequency; λ
cfs: wavelength at central frequency in free space)
NFDTD 18X18 NFF=1 (λcfs
/18)
NFDTD 18X18 NFF=2 (λcfs
/9)
NFDTD 18X18 NFF=2.25 (λcfs
/8)
Figure 5.12: The TE mode frequency spectra of the wave vector k at Γ point, for the modellingof the cylindrical copper rods in square lattice. NFDTD simulation result with a low spatialresolution (dxNFDTD = λcfs
9 ) agrees with the high spatial resolution (dxNFDTD = λcfs
18 ).With a spatial resolution (dxNFDTD = λcfs
8 ) close to the minimum one (dxNFDTD = λcfs
6 ),spurious mode (marked by the dotted red circle) can be observed with a weaker energy levelthan a genuine mode (marked by the dashed orange circle). At other probe positions the spu-rious energy at this frequency is much weaker than a genuine mode, so the average energyat this frequency is much weaker than a genuine mode, which results in this spurious modecan be filtered out numerically as noise and will not affect the calculation of the dispersiondiagram.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 178
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Normalized Frequency (ωa/(2πc))
Mag
nitu
de o
f Mag
netic
Fie
ld H
z (A
/m)
Frequency Spectra for wave vector k from the center of the Brillioun Zone to Γ point (TE) (NFF: Normalized Central feeding frequency)
Yee’s FDTD 80X80 NFF=2.2 (λcfs
/36)
Yee’s FDTD 36X36 NFF=1 (λcfs
/36)
Figure 5.13: The TE mode frequency spectra of the wave vector k at Γ point in reciprocalspace of the cylindrical rods in square lattice simulated by the Yee’s FDTD with adequatespatial resolutions (dxFDTD = λcfs
36 ). Spurious mode (marked by the dotted red circle) canbe observed with a much weaker energy level than a genuine mode (marked by the dashedorange circle). As a result, this spurious mode can be filtered out numerically as noise andwill not affect the calculation of the dispersion diagram.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 179
0 0.5 1 1.5 2 2.50
2
4
6
8
10
12
Normalized Frequency (ωa/(2πc))
Mag
nitu
de o
f Mag
netic
Fie
ld H
z (A
/m)
Frequency Spectra for k vector on Γ point (TE) (NFF: Normalized Central feeding frequency)
Yee’s FDTD 80X80 NFF=2.2 (λcfs
/36)
Yee’s FDTD 80X80 NFF=2.5 (λcfs
/32)
Figure 5.14: The TE mode frequency spectra of the wave vector k at Γ point of the cylindricalrods in square lattice, simulated by the Yee’s FDTD method using reduced spatial resolutions.By reducing the spatial resolution from (dxFDTD = λcfs
36 ) to (dxFDTD = λcfs
32 ), which is closeto the required minimum value dxFDTD = λcfs
30 , spurious mode (marked by the dotted redcircle) can be observed with a weaker energy level than a genuine mode (marked by thedashed orange circle). Since averaging this energy at this frequency upon all the probespositions gives a much lower energy level than a genuine mode, this spurious mode can befiltered out numerically as noise and will not affect the calculation of the dispersion diagram.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 180
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
1
2
3
4
5
6
7
8
9
10
11
Normalized Frequency (ωa/(2πc))
Mag
nitu
de o
f Mag
netic
Fie
ld H
z (A
/m)
Frequency Spectra for k vector on Γ point (TE) (NFF: Normalized Central feeding frequency)
Yee’s FDTD 50X50 NFF=2 (λ
cfs/25)
Yee’s FDTD 36X36 NFF=1 (λcfs
/36)
Figure 5.15: The TE mode frequency spectra of the wave vector k at Γ point in reciprocalspace of the cylindrical rods in square lattice. The blue line plots the Yee’s FDTD resultswith a spatial resolution lower than the minimum spatial resolution. Consequently, spuriousmode (marked by the dotted red circle) can be observed with the blue line with a comparableor higher energy level than that of a genuine mode (marked by the dashed orange circle). Asa result, this spurious mode can not be filtered out numerically as noise and will result in awrong dispersion diagram. The purple line plots the Yee’s FDTD results with an adequatespatial resolution.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 181
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
5
10
15
20
25
30
Normalized Frequency (ωa/(2πc))
Mag
nitu
de o
f Ele
ctric
Fie
ld E
z (V
/m)
Frequency Spectra for wave vector k from the center of the Brillioun Zone to Γ point (TM) for the copper rods in square lattice
( NFF: Normalized Central feeding frequency; λ
cfs: wavelength at central frequency in free space)
Yee’s FDTD 80X80 NFF=1.0 (λcfs
/80)
Yee’s FDTD 60X60 NFF=2.0 (λcfs
/30)
NFDTD 18X18 NFF=2.0 (λcfs
/9)
Figure 5.16: The TM mode frequency spectra of the wave vector k at Γ point for the cylindri-cal rods in square lattice, modelled using the NFDTD and the Yee’s FDTD with different spa-tial resolutions. Yee’s FDTD result with spatial resolution of (dxFDTD = λcfs
80 ) is used as a ref-erence. In Yee’s simulation result with spatial resolution of dxFDTD = λcfs
30 , spurious energy(marked by the red dotted circle) is observed with an amplitude comparable with a genuinemode (marked by the orange dash circle). At the same probe position, NFDTD simulationprovides all the genuine modes with a much lower spatial resolution (dxNFDTD = λcfs
9 ).
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 182
The minimum spatial resolutions above the system tolerance for modelling the afore-
mentioned structures (figures 5.1 and 5.2) required by the NFDTD and the Yee’s FDTD
methods is summarized in Table 5.2. The minimum values vary from different models
and operating frequencies, etc. (see figure 5.17). FDTD simulations with finite or infinite
EBG structures must meet their corresponding requirements and a high spatial resolu-
tion is always preferred for the consideration of accuracy within the capability of the
computer resources.
Table 5.3 shows how the computer resources can be saved by using NFDTD with a
lower spatial resolution. (The data are from the NFDTD and FDTD simulations for one
k-vector on the dispersion diagram of the copper rods in triangular lattice, TE mode.)
Spatial Resolution(dx) Mesh Size for 1 cell ( adx )
Model NFDTD YEE’s FDTD NFDTD YEE’s FDTD
Square Lattice λcfs
6λcfs
306·f · 2πa
c 30 · f · 2πacTE Mode
Square Lattice λcfs
9λcfs
309·f · 2πa
c 30 · f · 2πacTM Mode
Triangular Lattice λcfs
6λcfs
286·f · 2πa
c 28 · f · 2πacTE Mode
Triangular Lattice λcfs
9λcfs
309·f · 2πa
c 30 · f · 2πacTM Mode
λcfs: Wavelength at central working frequency in free space.
a:lattice constant; f : working frequency; c: speed of light in vacuum.
Table 5.2: Comparison of the minimum spatial resolution required by NFDTD and the Yee’sFDTD when simulating the copper rods infinite EBGs in square lattice or triangular lattice.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 183
0 0.5 1 1.5 2 2.5 30
10
20
30
40
50
Normalized Frequency (ωa/2πc)
Min
imum
Spa
tial R
esol
utio
n
(cel
ls/fr
ee s
pace
wav
elen
gth)
Copper rods array (r/a=0.2) Triangular Lattice. TE polarization
Yee’s FDTDNFDTD
(a)
0 0.5 1 1.5 2 2.5 30
20
40
60
80
100
120
Dom
ain
size
(in
x−
dire
ctio
n)
fo
r th
e un
it ce
ll
Normalized frequency (Normalized Frequency (ωa/2πc))
Copper rods array (r/a=0.2) Triangular Lattice. TE polarization
Yee’s FDTDNFDTD
(b)
Figure 5.17: The minimum spatial resolution (a) and mesh size (b) required for the Yee’sFDTD and NFDTD algorithms for the triangular lattice TE mode unit cell simulation.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 184
Unit NFDTD Yee’s FDTD
Operating FrequencyNormalized frequency 3 3
(NFF)Mesh size
cell2 18 × 18 72 × 72(Minimum required)
Spatial resolutioncells per free-space
6 24wavelength
Frequency resolution Normalized frequency 0.0139 0.0139
dt second 1.7187e− 11 1.7187e− 11
Total number of iteration10360 10360
(nmax)Computer Memory Used
MB 6.87 11.1(maximum value)
Computation timesecond 15.1 42.3
(on average)Relative Error Rate
% 7.99 4.25(Maximum)
Relative Error Rate% 1.53 0.38
(Mean)Averaged Relative Error
% 2.09 0.74Rate (ARER)
Table 5.3: Comparison of the memory, the computational time used (on average) in calculat-ing one k vector in dispersion diagram, and the accuracy of the results using the Yee’s FDTDand the NFDTD method.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 185
5.1.5 Discussion and Conclusion
All comparisons demonstrate the efficiency of the NFDTD algorithm over the Yee’s FDTD
algorithm when modelling EBGs with curved inclusions.
When the spatial resolution in the simulation is adequate, the dispersion diagram is
convergent from both algorithms. With the spatial resolution being decreased gradually,
the error increases and spurious resonances start to occur. The spurious energy is seen
to increases gradually as the spatial resolution being decreased further, until its energy
is comparable to that of a genuine eigenmode and hence result in a spurious eigenmode
of the EBG. As a result, for each model a minimum spatial resolution is required and the
spatial resolution used in the simulation need to be high enough to avoid the spurious
modes violating the dispersion diagram.
In the unit cell approach, the size of the computational domain is small as only a single
EBG element is included in the simulation. Consequently the computational burden is
small for both FDTD schemes and a dense mesh is affordable by both schemes. The
approach of finding the minimum spatial resolution using both the Yee’s FDTD and the
nonorthogonal FDTD helps to achieve the following conclusions:
• Knowledge of the spurious eigenmodes can be obtained. When predicting the
spurious modes due to an inadequate spatial resolution before any simulation is run, it is
quite likely to relate the small physical dimension (dx) to a frequency component in the
higher frequency band. However, simulation results show that spurious modes always
appear in lower frequency bands (among the first few genuine modes) for the staircase
approximation. Higher frequency bands are ’noise-free’, with all the modes genuine or
all the genuine modes much stronger than the spurious ones.
• To simulate any finite bandgap structure in the real world, the spatial resolution
should be high enough to avoid spurious modes in the simulation results. So a pre-
knowledge about the minimum spatial resolution required is necessary. This knowledge
can be gained by the study of the corresponding infinite EBGs via the unit cell approach.
• The minimum spatial resolutions depends on the configurations of EBGs, the
mode calculation and the FDTD scheme. For the same model, the NFDTD algorithm
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 186
always requires a much lower minimum spatial resolution than the Yee’s algorithm.
• As can be seen in figure 5.17, the mesh size required in modelling one unit
cell increases (nearly) proportionally with the increase of the operating frequency in the
Yee’s FDTD simulation. Consequently, when simulating a finite sized bandgap structure
operating at a high frequency in the real world, substantial computer resources (memory
and CPU time) are required. These facts make it impossible to simulate electrically large
EBG structures using the Yee algorithm.
• Despite the requirement of the additional variables and calculations in the NFDTD
method, the computer resource (memory and CPU time) used in the simulation can still
be reduced compared to Yee’s scheme by the use of a much coarser grid.
• When the mesh size is restricted by the computer memory, NFDTD method can
always deal with high order modes with less numerical errors. In this way, NFDTD is
more robust in terms of high frequency response than the Yee’s FDTD scheme.
• Due to the late time instability of the NFDTD algorithm, the energy of the tem-
poral signal will increase dramatically after a certain physical time in the simulation.
Consequently, the temporal signatures used for frequency analysis in post processing
should be truncated. However, a limited physical length of the time period will reduce
the frequency resolution after the Fourier transformation is performed. In order to pro-
vide the required frequency resolution, the truncated signal must be long enough. As
a result, the high power numerical errors will be involved in the post process which
will increase noise in the frequency spectra. At this point, the proposed ADI-NFDTD is
expected to reduce the late time instability of NFDTD algorithm. The ADI-NFDTD sim-
ulation results shown in Chapter 4 demonstrate the expected improvements in lowering
the noise level and delaying the instability into a later time.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 187
5.2 NFDTD Modelling of the (Semi-)Finite EBG Structure
In this section, a semi-finite two-dimension EBG structure (finite in one direction and infi-
nite in another direction) is studied by the NFDTD and the conventional Yee algorithms.
The bandgaps of the EBG structure are determined by a transmission coefficient calcu-
lation and they are compared with analytical results calculated from the transfer-matrix
method (TMM)[47].
5.2.1 The EBG Model
The EBG, as shown in figure 5.18, consists of arrays of dielectric cylindrical scatterers.
These cylinders have a radius r = 0.48cm and a dielectric constant εr = 9. They form a
square lattice with a lattice constant (period) a = 1.27cm. These rods are infinitely loaded
in the x direction. In the y direction, there are 8 layers of elements. In the z direction,
the rods are infinitely long. The ratio of the radius r to the lattice constant a (r-a ratio) is
r/a = 0.378. Numerical simulations are performed to determine the bandgaps for both
the TE and TM polarization.
Figure 5.18: The dielectric cylinders with radius r = 0.48cm, and dielectric constant εr = 9,surrounded by air, forming a square lattice with a lattice constant a=1.27cm; Infinite in z-direction; Infinite in x-direction; Finite (8 elements) in y-direction;
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 188
5.2.2 FDTD Model
Since the rods in the EBGs are periodic and infinite in the x direction, the entire struc-
ture can be studied by investigating only one row of EBG cells and by applying peri-
odic boundary conditions (PBCs) at the ends of x direction. In the y direction, a Perfect
Matched Layer (PML) is used to terminate the computational domain. The model is il-
lustrated in figure 5.19.
Figure 5.19: The cut plane of the EBG structure consists of 8 by infinite dielectric cylindricalrods array in square lattice, modelled by the NFDTD method. The rods are assumed to beinfinitely long, so a two-dimension model is applied. In the direction which the 8 elementsaligned in, PML is used to terminate the computational domain. In the direction which thearray is infinitely loaded in, the unit cell approach is applied with PBCs used to terminatethe computational domain.
A wide band plane wave propagating along the y direction is excited from a line
source at one side of the EBG rods. The wave form is Gaussian pulse which covers fre-
quency range 0 ∼ 20GHz.
At the other side, temporal signal is collected and Fourier Transformation is per-
formed. Then, after calibration, the transmission coefficients which indicating the ex-
istence of bandgaps can be calculated.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 189
5.2.3 Mesh of the Unit cell
The meshing of the unit cell under various spatial resolutions are shown in figure 5.20
and figure 5.21.
Figure 5.20: The conformal mesh of the unit cell of the structure described in figure 5.18 inthe x-y cut plane with spatial resolution of 48X48 per unit cell.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 190
(a) (b)
5
2
5
3
5
4
5
5
(c) (d)
Figure 5.21: The conformal mesh of the unit cell of structure (figure 5.18) in the x-y cut planewith different spatial resolutions ((a) 26X26; (b) 16X16; (c) 12X12; (d) 10X10 per unit cell).
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 191
5.2.4 Simulation Results
The temporal responses at the probes are plotted in figure 5.22, figure 5.23 and figure
5.24.
0 1 2 3 4 5 6
x 104
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Time steps
Magnetic f
ield
Hz (
A/m
)
Temporal result (source signal after time step 2e4 is all-zeros)
source
probe 1
Figure 5.22: The NFDTD temporal results (using mesh 5.20) after a Gaussian pulse planewave excitation at one side of the EBG slab.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 192
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10-7
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Time (s)
Magnetic F
ield
Hz (
A/m
)
Mesh: mycircle542
smaller
source
allprobes1,1(1,1:19500)
(r = 8 cells; a=26 cells)
(a)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10-7
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Time (s)
Magnetic F
ield
Hz (
A/m
)
Mesh: mycircle546
smaller
source
allprobes1,1(1,:)
(r = 5 cells; a=16 cells)
(b)
Figure 5.23: The NFDTD temporal results with different spatial resolutions after a ModulatedGaussian pulse plane wave excitation at one side of the EBG slab. (a) Mesh 5.21(a) is used;(b) Mesh 5.21(b) is used. )
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 193
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10-7
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Time (s)
Magnetic F
ield
Hz (
A/m
)
Mesh: mycircle547
smaller
source
allprobes1,1(1,:)
(r = 4 cells; a=12 cells)
(a)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10-7
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Time (s)
Magnetic F
ield
Hz (
A/m
)
Mesh: mycircle548
smaller
source
allprobes1,1(1,:)
(r = 3 cells ; a=10 cells)
(b)
Figure 5.24: The NFDTD temporal results with different spatial resolutions after a ModulatedGaussian pulse plane wave excitation at one side of the EBG slab. (a) Mesh 5.21(c) is used;(b) Mesh 5.21(d) is used.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 194
The transmission coefficients versus frequency for TE polarization calculated by the
NFDTD algorithm are plotted in figure 5.25. It can be seen that when the mesh size for
an unit cell is greater than agrid = 12 cells, the results are convergent. The transmission
coefficient shows considerate reduction around 11GHz and 15.5GHz. The first stopband
is not seen because the EBG layer in the y direction is relatively too thin (in terms of
electrical length) for prohibiting the lower frequency (longer wavelength) waves. The
NFDTD results show good agreement with the the TMM results (figure 5.26) and the
Yee’s FDTD results (figure 5.27). Good agreement is also seen in the TM mode results (
see figure 5.28 and figure 5.29).
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 195
Figure 5.25: The transmission coefficients (TE mode) of the dielectric rods (figure 5.18) calcu-lated using the NFDTD method. r is the radius of the cylinder.
Figure 5.26: The theoretical transmission coefficients (TE mode) of the dielectric rods (figure5.18) predicted by the transfer-matrix technique [47].
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 196
Figure 5.27: The transmission coefficients (TE mode) of the EBGs consisting of dielectric rods(figure 5.18). (a) the Yee’s FDTD simulation results (results from Dr Yan Zhao) and (b) thecomparison of NFDTD and Yee’s FDTD results with similar spatial resolutions.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 197
Figure 5.28: The transmission coefficients (TM mode) of the dielectric rods (figure 5.18) cal-culated using the NFDTD method with different spatial resolutions. r is the radius of thecylinder.
Figure 5.29: The theoretical transmission coefficients (TM mode) of the dielectric rods (figure5.18) predicted by the transfer-matrix technique [47].
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 198
5.2.5 Discussion and Conclusion
Both the NFDTD and Yee’s FDTD results show good agreement with the analytical results
predicted by the transfer-matrix technique when the spatial resolution is large enough.
However, there exist small differences in the parameters used by these two algorithms.
As it can be seen from figure 5.30, the NFDTD model uses an exact parameter set
by the physical model. Since the discretization in Yee’s scheme uses uniform cells, to
accurately model a r-a ratio of 0.48/1.27 = 48/127, a resolution of at least agrid = 127
cells and rgrid = 48 cells are needed. This is a too heavy computational burden for FDTD
simulation. So, a similar r-a ratio is used to reduce the necessity of a high resolution.
a = 1.28 cm is used in Yee’s FDTD model to make a r-a ratio of 0.48/1.28 = 3/8. As a
result, agrid can be any integral multiple of 8, with rgrid being the corresponding integral
multiple of 3.
r_grid = 8
a_grid = 26
r_grid = 12
a’_grid = 32
r_grid =16
a_grid = 46
r_grid = 24
a’_grid = 64
r_grid = 4
a_grid = 12
r_grid = 6
a’_grid = 16
r_grid = 5
a_grid = 16
r_grid = 9
a’_grid = 24
r_grid = 3
a_grid = 10
r_grid = 3
a’_grid = 8
NFDTD
a = 1.27cm
Yee’s FDTD
a’=1.28cmYee’s FDTD:
r = 0.48cm;
a’ = 1.28cm;
r/a’ = 3/8;
NFDTD:
r = 0.48cm;
a = 1.27cm;
r/a = 48/127;
Figure 5.30: Comparison of the FDTD and NFDTD simulation parameters.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 199
This comparison shows another advantage of the NFDTD algorithm over the Yee
algorithm:
• The spatial increments (dx or dy) of the Yee algorithm in one simulation are
fixed. Even with the subgridding scheme, dx (or dy) in each subdomain is fixed. On the
other hand, the spatial increments of the NFDTD algorithm are flexile. As a result, even
when the inclusions of the modelled problems are orthogonal, the NFDTD algorithm can
model structures with any dimensions with any ratios more accurately, more easily and
by using a simpler and coarser mesh.
5.3 NFDTD Modelling of an EBG-like Waveguide
A two-dimension EBG-like structure with defects working as waveguide is modeled us-
ing both the NFDTD algorithm and the Yee’s FDTD algorithm. (15 × 16) dielectric rods
with relative permittivity of εr = 11.56 and r-a ratio r/a = 0.2 are aligned in square lat-
tice, with a bended pass on which the rods are missing. In the simulations, PML is used
to terminate the computational domain. The waveguide structure is working in bandgap
frequency of a corresponding non-defect EBGs with a sine wave excited as a point source
on the defected pass. The wave is seen propagating along the waveguide with almost
all energies constrained within the defected pass. A snapshot of the field distribution in
the TM mode from NFDTD algorithm is shown in figure 5.31. It is similar to the FDTD
result. However, the NFDTD algorithm saves a lot of computation time by using a mesh
size of (550 × 590) compared to the mesh size of (890 × 940) in the Yee’s FDTD method.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 200
Figure 5.31: The NFDTD field plot near the bend of the EBG-like waveguide.
5.4 EBG-like Negative Refractor Modelling
Another EBG-like structure consisting of infinite long metallic wires which are arranged
in a prism shape was studied (ref. Figure (6) in [153]) to investigate the negative refraction
effect of the PBG-like structure. The wires are of radius r = 0.63 cm and r/a = 0.36.
The wires are arranged in a prism shape with 10 elements in the base and vertical sides,
respectively. A sine wave is excited from a point source in the middle from the left side
of the prism. PML is used to terminate the simulation domain. The structure is modelled
using the NFDTD method with a feeding frequency of 6.8 GHz and 7.6 GHz. The positive
and negative refraction reported in [153] from Ansoft’s HFSS simulation is verified by the
NFDTD simulation (see figure 5.32).
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 201
(a)
(b)
Figure 5.32: The electric field plot from the prism-shaped EBG like structure consisting ofmetallic wires, simulated by NFDTD method. (Unit is in V/m.) (a) the positive refractionseen with feeding frequency of 6.8 GHz. (b) the negative refraction seen with feeding fre-quency of 7.6 GHz.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 202
(a)
(b)
Figure 5.33: Direction of the electric-field-propagation refraction from the prism-shaped PBGlike structure with positive refraction (a) at 6.8 GHz, and the negative refraction (b) at 7.6 GHz(Unit is in V/m), simulated by Ansoft HFSS. [153]
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 203
5.5 ADI-NFDTD Modelling of the EBG Unit Cell
In this section, the performance of the proposed ADI-NFDTD method is demonstrated by
the modelling of an EBG unit cell ( figure 5.1) in comparison with the NFDTD method. A
reduction in the late time instability is demonstrated by the ADI-NFDTD method. Con-
sequently the ADI-NFDTD results are with a higher frequency resolution and a lower
noise level. As a result, some modes that are very hard to detect by NFDTD modelling
can be found easily in the ADI-NFDTD simulation results.
5.5.1 NFDTD Limitations
In the first numerical model of chapter 4, a dispersion diagram of an infinite 2-D EBG
formed by metallic rods periodically loaded in free space is examined by the NFDTD
simulations. However, with some k vector, the unstable NFDTD results occurs earlier
than others. This means that in order to keep the minimum frequency resolution, more
errors are involved in the truncated temporal signals which leads to a higher noise level
in the frequency spectrum. As a result, some genuine modes are ’buried’ in the noise and
become very hard to detect.
Take the previous copper rods in square lattice TE mode simulation for example. Both
the physical and reciprocal lattices are plotted in figure 5.34. The dispersion diagram cal-
culated by NFDTD is also recalled and re-plotted in figure 5.35. In those simulations,
while calculating k points near k = ΓM , the time domain signals exhibit instability
quickly. Figure 5.37(a) shows the temporal signature at one probe position for calcualt-
ing k = k1. Figure 5.36 shows the frequency spectra from different probe positions. As
a consequence of the exponential power numerical error, it is hard to find the mode at
frequency 1.35(normalized) from the spectra. As is shown in figure 5.36, this frequency
peak is missing with most probe positions (blue ones), and only with few probe positions
(red one as an example), this genuine mode is strong enough to compete with the noise
and form a very small peak (emphasized by a green circle).
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 204
Figure 5.34: The unit cell of the EBGs (figure 5.1) and the Brillouin Zone in the reciprocallattice with k = k1 shown by a red vector.
0
0.5
1
1.5
2
X: 23Y: 1.356
Fre
quen
cy (ω
a/(2
πc))
Copper rods (r/a=0.2) in Square Lattice
Γ X M k1 Γ
Figure 5.35: The dispersion diagram of the square lattice EBG, TE mode results, calculatedusing the NFDTD methods.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
50
100
150
Frequency Spectra at k1 point by NFDTD
Normalized Frequency (ωa/(2πc))
Mag
netic
Fie
ld H
z
Figure 5.36: The frequency spectra of different probes when calculating k = k1. It is hard todetect the eigenfrequency as a peak at frequency 1.356.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 205
5.5.2 Comparison of the ADI-NFDTD and NFDTD Simulation Results
A simulation is carried out by the ADI-NFDTD method for the same problem using the
same mesh file and the same time step dt. Temporal and frequency results are compared
with the NFDTD results (figure 5.37 and figure 5.38). As it can be seen, the resemblance of
the first 0.5 picosecond signals verifies the ADI-NFDTD algorithm. After that, the energy
of the ADI-NFDTD signal reduces gradually while that of the NFDTD signal increases
dramatically. After the FFT, the ADI-NFDTD frequency spectra show a lower noise level
and mode of 1.36 (emphasized by the purple circle in figure 5.38) is clearly shown.
0 2 4 6 8 10 12 14 16
x 10−8
−50
0
50
100
Time (s)
Mag
netic
Fie
ld H
z (A
/m)
Temporal signal at one Probe Point at k=k1 Point
Probe 1 by NFDTDProbe 1 by ADI−NFDTD
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10−8
−2
−1
0
1
2
Time (s)
Mag
netic
Fie
ld H
z (A
/m)
Temporal signal at one Probe Point at k=k1 Point
Probe 1 by NFDTDProbe 1 by ADI−NFDTD
(b)
Figure 5.37: The temporal response at one probe position when calculating k = k1. (a)Thecomparison showing the reduction of the late time instability by ADI-NFDTD. (b)The en-larged view of the first 0.5 picosecond signals, which shows the resemblance of the ADI-NFDTD results and the NFDTD results.
Chapter 5 NFDTD and ADI-NFDTD modelling of EBG Structures 206
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
50
100
150
200
Frequency Spectra at one k point by NFDTD and ADI−NFDTD
Normalized Frequency (ωa/(2πc))
Mag
netic
Fie
ld H
z
Probe of NFDTDProbe of ADI−NFDTD
Figure 5.38: The comparison of the frequency spectra in calculating k = k1 using the con-ventional NFDTD and the ADI-NFDTD. The NFDTD spectra which best indicates the modeat 1.36 is plotted in red line. The ADI-NFDTD spectra with an overall low noise level andclearly showing the mode at 1.36 is plotted in green line.
5.5.3 Discussion and Conclusion
In this section, the advantage of the ADI-NFDTD scheme over the conventional NFDTD
scheme in calculating the dispersion diagram of an EBG by means of the unit cell ap-
proach is demonstrated. When modelling the unit cell using the periodic boundary con-
dition based on different wave number k, the temporal responses become unstable much
earlier based on some k than others. This will cause an inadequate length of the time
domain result, a high noise level in the result, or both. As a consequence, some peaks
in the frequency spectra which stand for the corresponding eigenmodes of the EBGs are
very hard or impossible to detect.
To addresse this issue, the proposed ADI-NFDTD algorithm is used to model the
same problem. Much longer stable temporal results are achieved. Due to the longer
temporal result, the FFT is performed with a higher resolution. Because the high energy
unstable numerical error is avoided, the noise level in the frequency spectra is very low
and every eigenmode is clearly indicated by a distinct peak.
Chapter 6
Conclusions and Future work
6.1 Summary
Electromagnetic bandgap (EBG) structures have attracted world wide attention in the
electromagnetics and antenna communities due to their ability to guide and efficiently
control electromagnetic waves. Among a variety of numerical methods being applied to
study wave propagation and the bandgap properties of the EBGs, the Finite-Difference
Time-Domain (FDTD) method is popular because of its ability to deal with wideband
simulations from a time domain method and its ability to model complex structures. In
this thesis, the basics of EBG structures and the numerical methods for modelling EBGs
were reviewed in Chapter Two. Then the basics of the conventional Yee’s FDTD and the
techniques in relate to the modelling of the EBGs were presented in Chapter Three.
Two main aims were targeted in this study:
• To compare the numerical efficiency and accuracy of the nonorthogonal FDTD
method (NFDTD) and the Yee algorithm in modelling EBG structures.
• To modify the NFDTD algorithm into a more stable, efficient and accurate algo-
rithm.
The author’s main contributions with regard to these two aims start from Chapter
Four.
207
Chapter 6 Conclusions and Future work 208
In Chapter Four, the ADI-FDTD method is extended into the curvilinear system and
a novel ADI-NFDTD is proposed.
• The formulae of the ADI-NFDTD method (including the ADI-NFDTD incorpo-
rating the PBCs) are derived.
• Numerical simulation validates that in the ADI-NFDTD method, the CFL con-
dition of the FDTD method is removed. The computational efficiency may be increased
by using a larger dt in the ADI-NFDTD scheme than the conventional NFDTD scheme.
• Numerical simulation shows that the inherent late time instability of the NFDTD
method is largely reduced by the use of the ADI method. The ADI-NFDTD is still not an
unconditionally stable scheme. However, the ADI-NFDTD scheme is stable over a much
longer period of time.
In Chapter Five, the efficiency of NFDTD over FDTD when modelling EBGs with
curved inclusions is demonstrated in the numerical simulations:
• Both Yee’s FDTD and the NFDTD algorithm require a certain minimum spatial
resolution when modelling EBGs. To simulate a finite bandgap structure, a pilot simu-
lation can be run to find out the minimum spatial resolution required by modelling the
corresponding infinite EBGs with the unit cell approach. The minimum spatial resolution
required by the NFDTD algorithm is always much lower than that required by the Yee
algorithm.
• To reduce the late time instability of the NFDTD algorithm, the proposed ADI-
NFDTD scheme is applied in the modelling of the EBG unit cell. The ADI-NFDTD simu-
lation results demonstrate the expected reduction in the late time stability of the NFDTD
algorithm, including the delay of the numerical instability into a later time, an increase
of the frequency resolution and reduction of the noise floor in the frequency spectra.
• However, there are a few factors restricting the improvement in the computa-
tional efficiency of the ADI-NFDTD algorithm. Firstly, matrix calculation is introduced
in the ADI-NFDTD scheme and results in a more complex calculation scheme than the
conventional NFDTD. Secondly, the increase of the time step will result in a decrease in
Chapter 6 Conclusions and Future work 209
numerical accuracy. Therefore, in ADI-NFDTD, the time step should be carefully cho-
sen for the combined consideration of efficiency and accuracy. The choice of the NFDTD
mesh is also a factor which affects the numerical efficiency. With meshes locally fine and
distorted, the numerical efficiency improvement using the ADI-NFDTD is expected to be
more significant.
6.2 Future Work
6.2.1 Further Numerical Validations
So far in this work, the NFDTD method and the ADI-NFDTD method have demonstrated
their feasibility and advantage in modelling the EBG structures. Simulation criteria have
been found and the in-house NFDTD/ADI-NFDTD codes have been verified by some
pilot EBGs simulations. Novel EBG structures can be designed with the help of these
simulation tools. Three dimensional EBGs can also be modelled as further validations of
the NFDTD and the ADI-NFDTD method.
6.2.2 Enhanced NFDTD/ADI-NFDTD Algorithms
Improvements can also be made in terms of the NFDTD/ADI-NFDTD algorithms and
the associated programming.
• Enhancement in the NFDTD Method
The frequency and spatial dispersive FDTD is successfully developed for modelling
EBGs with frequency and spatial dispersions [12, 75]. At present, the method is based on
Cartesian coordinates. It will be interesting to extend it into the generalized curvilinear
Chapter 6 Conclusions and Future work 210
coordinate system and hence aim to achieve a frequency and spacial dispersive NFDTD
method.
• Extending the Two-Dimensional ADI-NFDTD Method into Three Dimensions
The ADI-NFDTD presented in this thesis is two-dimensional only. The following
derivations illustrate the difficulties in the attempts of extending the two-dimensional
ADI-NFDTD into three dimensions.
In Chapter 4, the partial differential maxwell’s equations (4.67)-(4.72) are written in
two procedures based on the ADI approximation. For simplicity, the first procedure is
presented here as an example. Assuming an isotropic medium with permittivity ε, the
three-dimensional ADI-NFDTD equation based on the updating equation for Ex is (de-
tailed derivation can be found in Appendix C):
−Coefxx1En+ 1
2x (i +
12, j, k − 1) + (1 + Coefxx2)Exn+ 1
2 (i +12, j, k)
− Coefxx3En+ 1
2x (i +
12, j, k + 1)
− Coefxy1En+ 1
2y (i, j, k − 1
2) + Coefxy1E
n+ 12
y (i + 1, j, k − 12)
+ Coefxy2En+ 1
2y (i, j, k +
12) − Coefxy2E
n+ 12
y (i + 1, j, k +12)
− Coefxz1En+ 1
2z (i +
12, j − 1
2, k − 1
2) + Coefxz1E
n+ 12
z (i +12, j +
12, k − 1
2)
+ Coefxz2En+ 1
2z (i +
12, j − 1
2, k +
12) − Coefxz2E
n+ 12
z (i +12, j +
12, k +
12)
=6∑
i=1
Sni (i, j, k) (6.1)
where the expressions for Si(i = 1..6) and the coefficients Coefxxp (p = 1, 2, 3) and
Coefxmp (m = y, z; p = 1, 2) can be found in Appendix C. There are two other up-
dating equations presented in a similar form relating to En+ 1
2x , E
n+ 12
y and En+ 1
2z . It is non-
trivial to solve these equations because a lot more variables are coupled to each other
compared with the two-dimensional ADI-NFDTD or the three-dimensional ADI-FDTD.
Chapter 6 Conclusions and Future work 211
Consequently, the feasibility of solving these equations as well as the necessary approxi-
mations need to be further studied.
• Develop the ADI-NFDTD Method Base on Douvalis NFDTD Scheme
The ADI-NFDTD method presented in this thesis is based on the widely used Lee’s
NFDTD algorithm. However, it has been demonstrated by numerical simulation and
in theory ( Appendix B), the Douvalis’ NFDTD algorithm presents better late time nu-
merical stability. So the ADI-NFDTD based on the Douvalis’ NFDTD algorithm (ADI-
DNFDTD) is expected to show better numerical performance.
• Development of the Envelop ADI-NFDTD Method
Since unconditional numerical stability of the ADI-FDTD for the full three-dimensional
case was derived [7, 22, 147], there was a revival of interest in the use and the develop-
ment of the ADI-FDTD. The Envelop ADI-FDTD is one of its extensions [156]. In the
Envelop ADI-FDTD, the envelope technique was coupled with the conventional ADI-
FDTD. It has demonstrated in simulation results, the superior performance over the ADI-
FDTD in terms of numerical accuracy. More to the point, this method retains the same
level of complexity in terms of the matrix calculations as the conventional ADI-FDTD.
As a result, this method could be a candidate to improve the numerical accuracy of the
ADI-NFDTD scheme.
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